That’s where the real game begins: finding out the in-game physics. It’s just like doing real physics except it’s cheaper, because you only need the game and not something like a giant particle accelerator.

For this experiment, I am going to be looking at *Super Smash Bros. Ultimate* on the Nintendo Switch. The goal of the game is to fight other fictional characters. But you can choose just about any of the Nintendo characters from previous games. That means you can have a battle between Pac Man and Mario.

I’ve done this videogame stuff before—for example, the physics of *Angry Birds*. *Super Smash Bros. Ultimate,* however, offers something extra fun. It has a training mode where you can test out two characters’ different moves. The best part is that you can control a virtual camera and have it stay zoomed out. This makes it easier to collect motion data. They also included some grid lines in the background as a measurement scale. It’s as if the developers created this game just for me.

I will start with a seemingly simple but important move: jumping straight up. In this case I am going to use the Captain Falcon character, for no particular reason. Screen capturing seems to be disabled for this game, so I had to record the actual TV with my phone. It seems to work out just fine though. After that, I can use video analysis (Tracker Video Analysis) to get a nice plot of the position as a function of time for Captain Falcon during his jump. This is what I get.

Rhett Allain

So, what can we get from this plot? First a quick note: I am going to assume the distance units for the grid are in meters. Why? Well, that would put the height of Captain Falcon at around 1.9 meters (around 6.2 feet), and that seems reasonable. Actually, I guess I should point out that there are three important things about jumping in any video analysis.

- The time scale (is the video running in real time or slow motion)
- The distance scale
- The vertical acceleration

If you jump on Earth (in real life), you know that time is real time (hopefully), and you know the distance scale. Once you get off the ground, there is only the gravitational force pulling you down. This means that you will have a vertical acceleration of -9.8 m/s^{2}. It doesn’t matter what your starting speed is—you will have the same vertical acceleration assuming air resistance is small and ignorable.

Now, when analyzing the motion in a video you have to make a choice. I can pick two of the above quantities (time, distance, acceleration) and then solve for the third quantity. For most videogames, I would assume that the time scale is real time and that it takes place on Earth with an acceleration of -9.8 m/s^{2}. I could then solve for the distance scale to see how big (or small) things are.

For this game, however, I am going to use the distance scale provided. Why not? If I also assume it runs in real time, that means I can solve for the vertical acceleration. But here we see there is a problem. If the character had a constant vertical acceleration, his position-time graph would be a parabola—but it’s not. Of course that never stopped me before, and it won’t stop me now. In the plot above you can see a data fit to just part of the data, the part that does look parabolic. From this fitting equation, I get a vertical acceleration of -5.69 m/s^{2}. OK, I can work with that.

Also from the position plot, it looks like Captain Falcon reaches some constant speed on the way down (but not on the way up). By fitting a linear function to this part of the data, I get a falling speed of 9.47 m/s. Some other data showed the jump was up to a height of 4.2 meters above from where he started.

What about a “double jump”? This is a classic videogame jump move, in which the character jumps a second time while in the air. Here is the position plot for something like that.

Rhett Allain

The upward motion is fairly complicated, so I will just leave that alone for now. However, we do get one really important piece of information from this jump. The character moves at a mostly constant speed on the way down. In this case, I get a speed of 10.9 m/s (which is very close to the 9.47 m/s from the previous jump).

OK, one more jump. How about a running jump? You know, where the character jumps both up and horizontally as though jumping over something. Here is the plot:

Rhett Allain

Let’s go over the important parts:

- This is a plot of the motion in both the horizontal and vertical directions at the same time.
- For the horizontal motion, he was running at a speed of 14.3 m/s before the jump. During the jump his speed was 6.95 m/s.
- Yes, he SHOULD have a constant horizontal speed during the jump—at least if this was subject to real-world physics (but it’s not).
- I’m not sure why his horizontal speed decreased at the beginning of the jump.
- Notice that in the vertical direction, the guy moves up and down as he runs. I just think that’s cool.
- The vertical motion seems similar to the case where he jumps straight up. On the way down, he moves at a constant velocity (I get about 8.9 m/s).
- For this jump, his maximum height is still about the same as when he jumped straight up.

I just added the “falcon” to the jump because it sounds awesome. But here is the important part of physics: You don’t really understand something unless you can model it. Yes, models. Science is all about models, so let’s make one. Of course I’m a big fan of numerical models in Python, so that’s what I will do.

But where do you start? Start with stuff you know. Make a model of a jumping human on Earth with real gravity. That’s not too hard. Here is what that looks like (with the code). Just to be clear, this is NOT the falcon jump. Click Play to run it again and the “pencil” to view and edit the code.

Now that I’ve got something working, I can start changing the rules. Here are the things I played with:

- Initial velocity—including a short “boost velocity at the beginning of the jump.”
- The vertical acceleration. I tried values of 50, 60, 70, and 100 m/s
^{2}(oh, and many others). - Falling terminal velocity.

In the end, I changed things up quite a bit from my conclusions based on the video analysis. Here is the rule that I have for a jumping Captain Falcon:

- Starts with an initial constant vertical velocity (88 m/s) until Falcon reaches 2 meters.
- A slightly lower constant vertical velocity (25 m/s) until a height of 3 meters.
- After that, a vertical acceleration of -60 m/s
^{2}with an initial velocity of 15 m/s. - Once the Falcon starts falling down, he has a vertical velocity of -4 m/s until he gets to a height of 4 meters.
- For the rest of the way down, he has a constant vertical velocity of -10 m/s.

Yes, that seems like a crazy model, but when I run it I get the following plot (along with data from the video analysis).

Rhett Allain

Check that OUT! That’s as close as I can get it. Here is the code for the graph (if you need it). But of course this isn’t the end. There is never an end to real science. Just because I have a model that works for THIS jump, I still need to see if the same rules work for other jumps. I will leave the test of other jumps to you as a homework assignment.

Honestly, this is what I love about physics analysis of videogames. It’s just like real science, but nowhere near as expensive.

But what about those impacts? I don’t mind watching football, but there’s no way I’m going to take one of those hits. When I see some of these epic tackles, I can’t help but think about physics.

There are some great tools available so that we can analyze the physics of a football hit. Really, we have everything. the masses of individual players? Yup—just search the roster and you can look these things up. Video analysis tools? Again, yes. Personally, I really like Tracker Video Analysis. There’s just two more things we need for a full analysis. I need the video frame rate, but this is trivial. Although some impacts are replayed in slow motion, they are also shown in real time. What about a distance scale? Oh wait! It’s right there on the field with the yard lines. We are all set.

Let’s start with a collision. I just did a search for “biggest football hits” and quickly found one that would work. In this case, I am going to look at the Clemson vs. Syracuse game from 2017. The play has Clemson wide receiver Trevion Thompson (205 lbs) tackled by Parris Bennett (216 lbs) from Syracuse.

I like this collision for two reasons. First, the motion is mostly up and down the field and not side to side. This means that the camera gets a side view of the motion and they line up with the yard lines for easy measurement. Second, Bennett actually lifts Thompson up and pushes him back. That looks cool.

Now, there is one small problem. The camera zooms in and pans. This means that the location of the origin changes with respect to the view of the camera and the pixel-to-meter ratio changes. Fortunately Tracker Video Analysis has a nice method to account for this camera motion—the calibration point pairs. Basically, you set the scale of the video and then track two points on the background. The app then adjusts the virtual coordinate system so you can mark the real locations of the two players. Here is the resulting view from that transformation.

Next, I just need to mark the location of both players in each frame. Here’s what I get.

Since these are position-time plots, the slope of the line will give the velocity of the player. Here is what I get. I’m using units of meters per second because I’m not a barbarian.

- Initial velocity for Bennett = 6.05 m/s
- Initial velocity for Thompson = -1.33 m/s
- Final velocity for Bennett and Thompson = 3.03 m/s

Great. But what can we do with these values? The answer is “more physics.” Let’s start with the momentum principle. This says that the total force on an object (or football player) is equal to the rate of change of momentum. But what is momentum? It’s the product of mass and velocity (represented by the symbol p). In fact, I can write the momentum principle in terms of the change in momentum like this:

Yes, momentum and the net force are both vectors. In this case we are just dealing with one dimension so it’s not really a big deal.

There is another key physics idea here: forces are an interaction between two things. When Bennett hits Thompson, it’s not just a one-way force. In fact, Thompson pushes back on Bennett with an equal and opposite force. It’s not about Thompson, it’s about the nature of forces. If Bennett hit a brick wall, the wall would still push back with the same force. During this interaction (also known as a hit) the two players have equal and opposite forces—but they also have the same time interval. Bennett can’t push on Thompson for a different amount of time that Thompson pushes on Bennett. Since they have the opposite force for the same time, this means that they have opposite changes in momentum. Or better yet, the total momentum before the collision is equal to the total momentum after the collision. This is the conservation of momentum.

OK, let’s write this as a one-dimensional equation in the direction of motion. I’m going to call the direction that Thompson starts off moving as the negative direction. That means that the conservation of momentum can be written as:

I have the values for the initial and final velocities as well as the masses (convert to kilograms). I can check this. Based on the data, the initial momentum (remember one of them is negative) would be 469.5 kgᐧm/s and the final momentum would be 579.1 kgᐧm/s. Yes, those aren’t quite the same but they aren’t super far apart. Here are the calculations (and more). You can change the numbers and press “play” to recalculate for fun (and you should). Yes, python is an awesome calculator.

So, why isn’t momentum conserved? It’s not conserved because there is another external force that’s not accounted for. No, it’s not the gravitational force. Although there is indeed a gravitational force pulling down, there is also an upward pushing force from the ground. The missing force is a sideways pushing force from the ground (a frictional force) due to Bennett pushing forward as he collides with Thompson.

I can approximate the contact time at around 0.2 seconds, then the external force on the system would have to be 548 Newtons (that’s 123 pounds for barbarians). But wait! This is the force on a system consisting of both players. If we just look at Bennett, he has to push on the ground with a greater force to compensate for the force that Thompson pushes on him. I can find that Thompson force by looking at his change in momentum in that same time interval. This gives a leg-pushing force of 2577 Newtons. Yes, that seems crazy high but it’s less than three times his body weight. Humans can easily achieve those super high forces for short times (see running as an example).

If you want some homework, here is something to try. Find another collision—one in which both players are off the ground during the collision. If both humans are off the ground then there would be no forward force. Momentum should be conserved. See if the numbers work. It will be fun.

They were right. Airbags are explosives and airbags save lives—but it’s still a crazy idea.

The original idea for the automotive airbag dates back to the early 1950s. It wasn’t exactly an explosive-powered device. It involved a compressed gas that would release to fill a type of bladder. This design didn’t work very well—it wasn’t fast enough. It turns out the only way to get an airbag to inflate fast enough to be useful is with an explosive. OK, technically it’s a chemical reaction that produces gas to fill the bag—but that’s essentially an explosion.

So, just how fast does the airbag need to inflate? Let’s do a rough estimation to get the minimum inflation time.

Here is the situation. It’s a nice day for a picnic. Let’s go on a picnic, shall we? I will bring some food and you can bring the blanket. Hop in the car and I will drive. Driving along and all of sudden—a deer in the road! Swerve! Crash into a tree. Blam.

The car was traveling only 35 miles per hour (15.6 meters per second) and when it crashes into the tree, it stops instantly (which isn’t true—but close enough). There is one other important thing to estimate, the distance from the driver to the steering wheel. I’ll be generous and put this at a value of 0.5 meter; it’s probably lower than that.

Here is what happens. The human is traveling at 15.6 m/s inside a car that just stopped instantly. The air bag has to deploy before the human collides with the stopped steering wheel. Now for some physics. If we assume no forces on the human (forget about the seatbelt for now), the human will have a constant velocity. With this constant velocity, we can calculate the time to impact using just the definition of average velocity in one dimension:

Using the values of 15.6 m/s for the velocity and a distance of 0.5 meters, I get a collision time of 0.032 seconds. That is a super short time—like the blink of an eye. OK, actually it is less than the “blink of an eye” which seems to take around 0.1 seconds. But wait! It’s even worse. This calculated time is how long it would take for the human to hit the steering wheel. But the idea is to have the human hit an airbag instead. That means that not only does the airbag have to inflate in this time, but it also decreases the distance the person can travel. Let’s say the distance is cut in half—that also cuts the inflation time in half to about 0.016 seconds.

Try blowing up a balloon in under a tenth of a second. It’s pretty much impossible. Now try doing the same thing for a larger bag in even less time. You can’t do that with air. The only answer is an explosion that produces a gas to fill the airbag.

It’s even crazier than you think. Let’s imagine what has to happen for this collision (which is just at 35 mph).

- The car hits the tree.
- Something in the car has to detect a collision. There has to be an accelerometer that says “oh hey—I think we are crashing.”
- This sensor then sends a signal to the airbag. The airbag then ignites the “explosive” inside.
- The explosion expands and inflates the bag.

All of that has to happen before the human hits the bag. Crazy. OK, it’s true the seatbelt also slows the human down during the collision. This will give the airbag a little more time to deploy. This is also why you should wear your seatbelt—even if your car has an airbag. Or better yet, just watch out for deer on the road and drive carefully.

Actually, the stuff I’m going to look at in this post doesn’t really involve any major plot points. It’s not like I’m going to reveal that Darth Vader is Luke’s father (oops—slight spoiler alert if you haven’t seen *The Empire Strikes Back*). OK, here is your last chance to bail on this physics post. You have been warned.

Have you ever noticed how some roads (particularly those tight curves on the exit ramps) have roads that aren’t level? There’s a reason to add a bank to these turns—they make it easier for cars to turn without crashing. Why?

Let me start with the definition of acceleration. An object accelerates when it changes its velocity. If we look at some short time interval (Δt), then the acceleration during this time interval would be defined as:

Rhett Allain

But what the heck are those arrows above the “a” and “v”? Those arrows are there to indicate that the acceleration and the velocity are vector quantities. A vector is a type of variable that has more than just one “piece” of information. A velocity vector could have three “parts”—a component for each dimension (since we live in 3 dimensions). The same is true for acceleration—it has three components. So, it doesn’t just matter about the total speed but also the direction of that speed (which we call velocity).

The acceleration depends on the CHANGE in velocity—but the velocity is a vector. This means that just by changing the direction of the velocity (also known as turning) you have an acceleration. Moving in a circle at a constant speed is an acceleration. But acceleration is a vector too! The direction of the acceleration vector for an object moving in a circle is towards the center of that circle.

Just to make sure everything is clear, here is a diagram showing the top view of a car driving in a circle at a constant speed. You can see the car at two different times with velocities in different directions. I also put an arrow showing the direction of the acceleration vector.

Rhett Allain

But how do you make an object accelerate? In order to have an acceleration, you have to have a net force in the direction of the acceleration. So, if I have a car turning in a circle there has to be a force pushing towards the center of the circle (since that is the direction of the acceleration).

Here are two different forces that could cause a car to move in a circle. In both of these diagrams, the car is driving towards the viewer and turning to the left of the screen.

Rhett Allain

For the car on the flat curve (the one on the left), there is one force pushing towards the center of the circular path—that is the frictional force. You need friction between the tires and the road to get the car to turn. That’s why some people crash on icy roads—there’s not enough friction to turn.

For the car on the banked turn, there is one big difference—it’s that force labeled F_{N}. This is the force that the ground pushes up on the car and it’s perpendicular to the ground (that’s why there is an “N” for normal). In this banked turn, the ground force does two things. First, it pushes up to counteract the downward gravitational force. Second, it has a component that pushes in the direction of the center of the circle. So this ground force is what causes the car to have an acceleration. If you get the angle and the speed of the car just right, you don’t even need a frictional force to turn the car. It doesn’t matter if the road is wet, icy, or dry—it can still turn with a banked road.

We have this force-motion model in physics. It says that the net force on an object is equal to the product of mass and acceleration. But what is a force? A force is an interaction between two objects—like when you push on the wall, or when the Earth’s has a gravitational interaction with the moon. However, there is one thing about this idea—it only works if you view things from a non-accelerating reference frame (an inertial reference frame).

What does this have to do with Star Wars or turning cars? Well, suppose you are in a car and that car makes a turn. Let’s say the car turns to the left and you are sitting in the passenger seat. How does it make you feel? It feels like you are getting pushed into the door, doesn’t it? It’s some invisible force from this turn—except it’s not. There is no force pushing you away from the turn. Instead, this is the side of the car pushing you IN to the turn. But since you are in the car and the car is accelerating, you have put yourself in an accelerating reference frame and the force-motion model doesn’t work.

This is where we get to use fake forces. If you want to take an accelerating reference frame and make it act like a non-accelerating frame you need to include fake forces. These fake forces are in the opposite direction of the acceleration of the frame and make it so the force-motion model works again. So that is a fake force pushing you into the door of the car during a turn. Some people call it the “centrifugal force”—and that is fine as long as you remember that it’t not real.

Back to the cars on the flat and banked curve. Let’s add something in the car—some fuzzy dice hanging from the mirror. When the car turns, there will be three forces on this fuzzy dice (in the reference frame of the accelerating car). In the car on the flat turn there will be the downward force from gravity, and the horizontal force from the fake force pushing away from the center of the circle. In order to make all the forces add up to zero, the string holding the dice must pull at an angle.

What about the dice in the car on a banked curve? There is still a fake force pushing away from the center of the circle. However, since the car is tilted the gravitational force is not “straight down” (with respect to the car). This means the fake force can cancel the sideways part of gravity and the dice hangs “straight down.” Here, maybe this diagram will help.

Rhett Allain

Notice that really the hanging dice are the same in both cars—it’s just the car orientation that’s different. From the perspective of the banked turning car, it feels the same as though gravity was pulling “down” but just a little bit harder. There is no “sideways” force.

There is another way to think about the inside of a turning car using the Equivalence Principle. Albert Einstien said that an accelerating reference frame is indistinguishable from a gravitational field. This means that if you are in a box with no windows and you feel your weight, this could be due to a gravitatational force OR because the box is accelerating.

With this idea, the inside of a turning car is the same as a gravitational field that is the sum of the Earth’s gravitational field and the fake field from the acceleration. A car turning on a flat road has a net equivalent field that is diagonal to the floor such that things get pushed to the outside of the turn. A car on a banked turn has the equivalent gravitational field pointed straight towards the floor so that you don’t move to the side, you just feel a little bit heavier.

We finally get to the scene in *Solo: A Star Wars Story*. In this part of the movie, it’s essentially a train robbery. But wait! This train is on an elevated rail and even better, it leans on the tracks when taking a turn. Han and Chewie are on the train during the turn but it’s too late. The storm troopers arrive. The storm troopers are prepared. They have magnetic boots so they won’t fall off the turning train. Chewbacca is not prepared. He almost falls off—but his BFF is there to save him (BFF means best friend forever).

Here is my artist rendition of this scene.

Rhett Allain

Now you see the problem, right? If they are on a turning train that is on a banked turn, “down” would be towards the floor of the train, not the real down. Han doesn’t even need to save Chewie—physics can save him. Oh, and the storm troopers don’t even need magnetic boots.

If the train was on a level and turning track, THEN Chewbacca could be falling out and the storm troopers would need magnetic boots. But not for a turn. OK, I know what you are saying. Yes, it is indeed possible that the train is on a banked track that is not designed for that speed. If the train was going too slow, then yes—the scene is exactly correct.

Don’t get me wrong, I still like the movie. I just like physics also.

Of course even if you don’t surf, there is still some cool physics involved in the act of surfing (let alone all the physics of wave formation). So, how does a rider stay moving with a huge wave like this? As with all motions, the key is to look at forces.

Here’s a diagram showing the forces on a surfer moving at a constant speed on a wave.

Remember that these forces all have to add up in a way to make zero net force. If there is a non-zero net force on the surfer then there would also be an acceleration (that is the nature of forces). But what are these three forces? The most obvious is the gravitational force. This is the downward pulling force that results from the mass of the surfer and the Earth interacting. It’s a cool force, but since neither the masses nor the distances between the objects change—this force is also constant.

The next force to consider is the drag force. As the surf board moves through the water, there is some resistance. This resistance is like a frictional force in the opposite direction of motion and parallel to the surface of the water. Notice that the water for this wave is not horizontal—since it’s, you know, a wave. I assume the magnitude of this drag force would increase with an increase in surfer speed.

The last force is the one I have labeled as “water.” Yes, I am cheating here. Technically the “drag” force is also from the water, but I didn’t know what else to call this. The important point is that this component of the force from the water is perpendicular to the surface of the water instead of parallel (like the drag force). It comes from a combination of buoyancy and a type of “lift” as the board moves through the water. This water force prevents the board from “sinking” into the wave. Add up all these forces and the surfer moves at a constant speed.

But wait: It gets more complicated. In order to see the problem, I will draw a slightly more detailed force diagram—but the surfer will just be a dot.

Consider the line parallel to the water. I will call this the x-axis. There are two forces pushing in this x-direction. There is the drag force and then there is a component of the gravitational force. As the wave gets steeper (greater value of θ) there is a larger component of the gravitational force in the direction down the wave. In terms of the gravitational force (mass multiplied by the gravitational field—g) and the wave angle, the following must be true for constant velocity.

So, a steeper wave means a greater “forward” pushing force which would speed up the surfer. But a greater speed probably means a greater drag force. This means that for a given wave angle, there is only one surfer speed that will give a particular velocity. That’s important because you probably want to be traveling at the same speed as the wave—that’s the whole point of this sport called “surfing.” Oh, I should point out that in this example the wave and the surfer are moving in different directions—the wave is moving horizontally, but the water in the wave moves in complicated ways.

Thus it seems there should be a particular spot on the wave for a surfer to “catch the wave.” If a surfer is moving too fast for a wave, there is always another option—turn. If a surfer moves perpendicular to the direction the wave moves, the speed in the direction of the wave is much lower. I guess (reminder: not an actual surfer) this is why they travel along the length of the wave. This is likely doubly true for wave that’s as giant and steep as this one in Portugal.

I still would be afraid to even be near this type of a wave.

In a recent episode, one of the large spaceships (the Navoo) rotates in order to create artificial gravity (that’s not really a spoiler). How about some questions and answers about this giant spinning spaceship?

Let me get right to it. You are probably somewhere near the surface of the Earth and there is a gravitational force between you and the Earth pulling you down. But here is the crazy part—you don’t really feel this gravitational force. Since the gravitational force pulls on all parts of your body, you don’t feel it. What you actually feel as “weight” is the force from the floor (or seat) pushing up on you. We call this the “apparent” weight.

Here is a case where you can see how this works. Go to an elevator and get in. Press the “up” button. When the door closes, the elevator starts to accelerate upwards. How do you feel? Anxious? Bored? Heavy? Yes, you feel a little bit heavier than normal—even if just for a short time. In order for your body to accelerate upwards, there must be an upward net force. This means the force of the floor pushing up must be greater than the weight pulling down. The actual weight doesn’t change, just your apparent weight. Like I said, you don’t really feel the gravitational force.

OK, get out of the elevator and get in a spaceship. It doesn’t matter if this spaceship is in orbit around a planet or traveling between planets. All that matters is that the motion is *only* due to a gravitational interaction (with the planet or with the sun). In that case, everyone inside will *only* have the gravitational force and they will feel weightless. So, how do you make “weight”?

One solution to this problem is for the spacecraft to accelerate. An accelerating spacecraft means the people will also accelerate and an accelerating person needs a force pushing on them The force pushing on the humans becomes the apparent weight—just like in an elevator.

This is the way most spacecraft create “weight” in *The Expanse*. They use their thrusters to accelerate so that they are either speeding up or slowing down. But either way, there is a force on the humans inside and they feel “weight.” If the spacecraft accelerates at 9.8 m/s^{2} (which is the same value as the gravitational force on the surface of the Earth) then the humans inside would feel just like they do on Earth.

But wait! There is another way to accelerate a human inside a spacecraft—make the human move in a circular path. Since acceleration is defined as the rate of change of velocity and velocity is a vector, just changing direction is an acceleration. The magnitude of the acceleration while traveling in a circle depends on both the speed and radius of the circle.

Since it’s a rotating object, it’s sometimes better to talk about the acceleration in terms of the angular velocity (ω) in radians per second. Oh, the direction of this acceleration (as a vector) is towards the center of the circle. This means you can make a rotating spacecraft in which the humans stand on the inside with their heads towards the center of rotation and they can feel “weight.”

Boom. This is exactly what happens with the Navoo in *The Expanse*. Oh, there is actually a plot point in that only the Navoo can create artificial gravity.

Great question. For all the previous episodes, the Navoo did not rotate. They created artificial gravity just like everyone else—they had rockets accelerate the spacecraft. Oh, you should note that often the spaceships are flying backwards because they don’t actually fly in space. Flying implies using air to create lift and that’s not what’s going on here.

If you want to go from point A to point B, here’s what you do. Step one: Turn on your thrusters and speed up your ship with an acceleration so that you have artificial gravity. Step two: Halfway there, turn the spaceship around so that it is facing opposite the direction of travel. Now turn on the thrusters again so that the ship slows down—yes, this is still an acceleration and you would still have artificial gravity.

Now for the problem with the Navoo—it’s just a small problem. Let me draw this ship as a giant cylinder.

Take a look at these two people (labeled A and B). For person A, they would experience fake gravity when the ship is increasing in speed or decreasing in speed—that time when the thrusters are on. But for person B, they would have “gravity” when the ship is rotating.

So, where is the bridge—the place where they control the spacecraft? The real answer is that they probably should have two separate bridges. There would be a bridge for “thruster-mode” and a bridge for “spinning-mode.” Here is the problem. In the show, it looks like there is just one bridge. But really—it’s just a small detail and it doesn’t really bother me. I’m just pointing it out for fun.

In the show, they show the Navoo rotating up to spinning speed. Here’s all the data I can get from this part of the episode (along with some dimensions of the Navoo).

- It takes 30 seconds for the Navoo to get up to spinning speed. OK, I get it. This might not be the actual time, but in the episode it seems like they show the whole spin up time. I’m going with 30 seconds.
- The Navoo has a radius of 0.25 kilometers and a length of 2 km.
- At the end of the spin up, they declare that they are at “one ten g”. I assume this is 1/10
^{th}g, but these are belters. Belters have this accent that is a lot like Eliza Doolittle before she was transformed into My Fair Lady. If you don’t get that joke, you should seriously watch that movie—it’s great.

Let’s start with the one ten g (I’m going to start talking like a belter). If the humans in the “drum” (the cylinder part of the ship) have an apparent weight of 0.1 g (0.98 m/s^{2}), then I can calculate the final angular velocity by using the radius of the drum.

With the values for the acceleration and radius, I get a rotation speed of 0.063 radians per second or 0.6 rpm (so not very fast). But note that the Navoo doesn’t need a super fast rotation rate since it is so huge. Just for fun, here are the size and rotation rates for some other science fiction spaceships.

If you want get this spaceship rotating at 0.063 rad/s in 30 seconds, the angular acceleration would be 0.0021 radians/s^{2}. Yes, that seems small—but remember, this is a huge ship. How much energy would it take to get this thing rotating and what about the power output? I will leave these as homework questions. You will have to estimate the mass and the moment of inertia for the Navoo (or you could assume it’s a hollow cylinder).

But wait! There’s more. We actually see the Navoo increasing in rotational speed in the show. I can use Tracker Video Analysis to get the angular position of the side of the ship as a function of time and from that find the angular acceleration. Here is the plot of angle vs. time.

Fitting a quadratic function to this data, I get an angular acceleration of 0.00038 rad/s^{2}. Yes, this is significantly smaller than the acceleration based on the 30 second time interval from the show—I’m actually surprised. I would have expected the angular acceleration from the visual of the rotating drum to be higher than calculated so that you could actually see it increasing in rotation speed.

OK, how about another homework question? With the angular acceleration from the video, how long would it take to get up to a rotation rate to produce 0.1 gs?

There is another view of the rotating spacecraft from inside the “drum.” The center of the Navoo is basically empty with all the people on the inside wall—but in this case there are a bunch of objects “floating” in the middle after some incident (I won’t say what happened). As the Navoo starts to rotate, you can see this “floating” stuff do two things. First, they appear to rotate because of the relative view from inside the rotating spacecraft. This is legit and what should actually happen (I love saying “actually happen” for science fiction stuff). Second, the inner debris starts “falling” towards the outer walls of the rotating drum. This part is wrong.

Why wouldn’t the debris “fall” under the influence of the artificial gravity? It wouldn’t fall because it’s not rotating. The debris doesn’t rotate around the axis of the ship because nothing pushes it to get it spinning. The rest of the stuff in the ship is connected to the floors and walls. There are rockets that increase the rotational speed of all of the stuff connected to the ship, but that doesn’t happen for the floating debris. No rotation means no artificial gravity.

Oh, but eventually this inner floating junk would fall. As the ship rotates, it will eventually push the air in the center to also spin. This spinning air will feel like a wind and the wind would slowly push this debris into the wall-floor (not sure what to call the inner side of the drum). But still, it’s a cool looking effect—even if not accurate.

Overall, this is a great show—*and* it gets most of the science right!

Perpetual motion—it’s fun to say that. For some people, perpetual motion machines hold the secret to everlasting free energy that will save the world. To them, it’s a machine that is just beyond our grasp. If only we could tweak our design just a little bit, it would work. To others (like me), perpetual motion machines are impossible—they don’t fit with our well-tested ideas of the conservation of energy. However, they can still make a fun puzzle, as you see above.

OK, so exactly what is a perpetual motion machine? The basic idea is that these machines can operate in some manner (say with a spinning wheel) forever without an energy source. That’s not so impossible, just not very easy (since machines always lose energy to friction). A better perpetual motion machine not only works forever, but also has a positive energy output (even if just a small amount) so that it can save the world—but like I said, these don’t exist.

Why don’t perpetual motion machines work? It’s all about energy. In short, energy is this quantity that we can calculate for some system. It turns out that because of the way we define energy, this quantity doesn’t change for a system that doesn’t have external interactions. That, in a nut shell, is energy conservation (where “conservation” means “does not change”). Energy isn’t really real—it’s just something that works. It works time and time again. It works so well that if we find a case in which it doesn’t work, we look for some type of missing energy (and we find it).

OK, so now we have a perpetual motion system. In theory, a “real” perpetual motion machine would have zero energy added to the system but still have an energy output—forever. How do we test this? The best way is to somehow isolate the machine so that we’re sure there are no energy inputs. That means blocking out light, radio waves, microwaves, magnetic fields, and thermal temperature differences. Don’t even let the thing shake. If you can completely isolate it and it *still* has an energy output, then you just changed everything we know about physics. Congratulations. You deserve some ice cream now.

Now for the puzzle. How does this particular machine work? It’s clearly not a *real* perpetual motion machine—there is some trick to make it work. I would love to play with this thing, but all I have is a video. If I could try some stuff, I’d like to see if it runs without lights on it. I would also see if I could measure any temperature differences. Remember, this thing doesn’t need much energy for a mere spinning wheel. If the friction is low enough, you really wouldn’t need to add much energy at all.

With that being said, I am going to make some guess as to how this works. Here are my ideas.

**Temperature Difference**. There is some type of tiny temperature difference between the top and bottom of the device. This means that there could either be a convection current or some type of expanding gas that can get this wheel to rotate. Another option would be to take this temperature difference to power a thermoelectric device (and here is how those work).

**Solar Powered**. Not real solar powered with a solar cell and a motor—that might be too obvious. Instead it could use solar power as a method for differential heating. Yes, this would be just like the previous idea unless it uses some type of actual solar cell (which I don’t see).

**A Frickin Battery**. Yes, this is a real possibility. The stupid wheel could just have a battery in there that keeps it spinning for long enough that no one can tell. I wouldn’t be surprised if it’s a battery with all the other things added as a distraction.

I guess we will never know how it works—unless I get my hands on that letter with the answer.

But if you think about it, this perpetual motion machine is a great example of the nature of science. The machine is real life, and we (the humans) are trying to figure out how it works. We come up with different ideas and then find some way to test if our ideas are legit. Suppose I think the sucker is solar powered. If that is indeed the case, I could cover the machine with a cloth and it should stop working. Of course we’d never really know if we were right about how it worked, but you’d know if you are wrong—if the “test” didn’t do anything.

So, it’s not just a puzzle—it’s an artistic expression of the nature of science.

A block has a mass of 1 kg and is placed on a vertical wall such that the coefficient of static friction is 0.5. With what magnitude force do you need to push on the block perpendicular to the wall to keep the block from falling?

Here, a picture will help.

Let’s do this. I will start with a force diagram. There are four forces acting on this block: downward gravitational force, the force from the push, a frictional force, and a normal force from the wall. As a force diagram, it would look like this.

Since this block is in equilibrium (at rest with zero acceleration), the net vector force must be zero Newtons. This means the net force in the *x* direction and the net force in the *y* direction must be zero. What forces are acting in the *y* direction? It’s just the gravitational force and the frictional force. Yes, the magnitude of the frictional force must be equal to the weight. The friction is in the vertical direction because it is parallel to the wall (which is vertical).

But how do you increase the friction force so that it’s enough to keep the block from falling? In a basic model of friction, the magnitude of this force depends on two things: the types of materials interacting and the force with which they’re pushed together. You can’t really change the types of material in the block and the wall—but you can control the force with which they’re pushed together. In the diagram above, this force is the normal force (labeled as N). In general, we can write this friction model as the following:

So the pushing force (*F* above) increases the normal force (*N*) and this normal force increases the friction force to balance the weight. I’ll skip the rest of the steps (you should do this for homework), but in the end, the minimum force (which uses the maximum friction) would be:

Does this answer even make sense? Yes. First, it has the right units (units of Newtons) since the coefficient of static friction is unitless and *m*g* is in Newtons. Second, this says that with a greater coefficient of friction, you don’t have to push so hard. That makes sense. But really, this problem is just a warm-up. How about something better?

Next question. What if I take the same block and push up at an angle? Like this:

Would I have to push harder or not as hard to keep the block up? Go ahead and make a prediction. OK, now let’s do it. Again, I will start with a force diagram. It’s pretty much the same as before except that pushing force is at an angle.

Except that it’s not the same. Some cool things happen. First, since the force is pushing at an angle the *x* component decreases. This means the normal force also decreases, which decreases the frictional force. However, that’s OK since the force now has a vertical component too. With this force diagram, I can again write down the sum of the forces in the *x* and *y* directions. I will skip most of the steps (so that you can do it for homework), but here is what I get for the magnitude of the force pushing at an angle (I also used the friction model from above).

That’s nice. Right? But again, we should check this equation. Does it have the correct units? Yes—the denominator of this expression just has μ_{s} and trig functions which do not have units. This means that overall the expression has units of Newtons. That’s good. What would happen if I push with an angle of zero degrees? Putting in zero for θ in this expression I get the same answer as the first question (pushing horizontal)—so that makes sense. Finally, what if I push straight up with θ equal to 90 degrees? This would give a force magnitude equal to the weight of the block—again, that seems reasonable.

But wait! What about the answer to the question about pushing horizontal compared to pushing at an angle? Which requires more force? As long as the angle is between 0 and 90 degrees, the stuff in the denominator will be bigger than just μ_{s} such that the force will be smaller at an angle. Is there a “best” angle that requires the least amount of force? Yes! Let’s find it.

Of course you could take the equation for the force as a function of θ above and take the derivative with respect to θ and then set it equal to zero. This would be your standard max-min problem from calculus class.

What if I just sort of use brute force to solve this problem? I can calculate the force required and a bunch of different angles and then just find the angle with the smallest force. Since I don’t actually want to do this by hand—I will use Python. And here is that program (so that you can play with it too).

Just push the Run button to run it and then you can go back and edit the code if you want to play with it. Don’t worry, you can’t really break anything. This is a fairly straightforward program. But the cool part is that there is a minimum pushing force at some angle. It’s easier to see with larger values for the coefficient of friction, so in the code above, I have it set at 0.6. With this value, a push angle of about 59.9° gives the lowest possible force. This force is lower than pushing it horizontal and lower than pushing straight up. In fact, every angle gives a lower force than pushing it horizontal.

What happens as you change the coefficient of friction? Just for fun, I made the program a little bit more complicated so that I can create this plot. (Here’s the code if you want it)

I don’t care what you think. This is cool. See, you thought it was a boring physics problem, but you were wrong. Oh, what is this good for in real life? It doesn’t matter. I still think it’s awesome.

Other than playing around with the code, I just have one homework question for you. Given the same values for mass and the coefficient of static friction and pushing angle—what is the *maximum* force you can push on the block before it slides *up*? Hint: you will need to change the direction of the frictional force. Also, you should try changing the angle and see what happens—you know … for fun.

Are superheroes real? Maybe. In this recently released video, a firefighter in Latvia catches a man falling past a window. Let me tell you something. I have a fairly reasonable understanding of physics and this catch looks close to being impossible—but it’s real.

Here is the situation (as far as I can tell). A dude is hanging on a window (actually, the falling human is only rumored to be a male) and then he falls. The firefighters were setting up a proper way to catch him, but it wasn’t ready. Of course the only solution is then to catch him as he falls. It seems the victim fell from one level above the firefighter. At least that’s what I’m going to assume. Now for some questions and answers.

This is a classic physics problem (I hope my students are paying attention). An object (or human) starts from rest and then begins to fall under the influence of the gravitational force. If the gravitational force is the only significant interaction on the human then that person will fall with a constant acceleration of 9.8 m/s^{2}. That means that for every second of free fall, the human’s speed will increase by 9.8 m/s (hint: 9.8 m/s is fairly fast—about 22 mph).

If I knew the time the human was falling, I could easily determine the speed since it increases a set amount every second. However, I can only approximate the distance the person falls. Of course that is only a small stumbling block for physics. In fact there is a kinematic equation that gives the speed of an object with a constant acceleration after a certain distance (you can also easily derive this with the definition of average velocity and acceleration). But if the object starts from rest and moves a distance y, then the final speed will be:

Yes, the greater the fall, the greater the speed. In this case, I’m just going to guess the distance at about 3 meters (it’s just a guess). That would put the speed of the faller (is that a real word) at about 7.7 m/s. Maybe it’s a little bit shorter fall at 2 meters—that would give a window-level speed of 6.3 m/s. Either way, it’s fast.

It doesn’t take a superhuman to fall but it might take superhuman strength to stop someone during a fall. The key here is the nature of forces. A net force on an object changes the motion of that object. In this case, there will be two forces acting on the falling human. First, there is the gravitational force pulling down. This force depends on the gravitational field (g = 9.8 Newtons per kilogram) and the mass of the human (which I don’t actually know). The second force is that of the firefighter pushing up during the catch. The total force (sum of these two forces) must be in the upward direction so that the change in motion is also up. This means the human (during the catch) will be slowing down. That’s what we want.

I can estimate the human’s mass, but what about that firefighter force? There are two basic ideas that deal with force and motion. First is the momentum principle. This is a relationship between force, momentum (product of mass and velocity) and time. The second is the work energy principle. This deals with forces, energy, and displacement. So it comes down to this. Do I want to estimate the *time* it takes to catch the human or do I want to estimate the *distance* over which the human was caught? I think I’ll go with distance and the work-energy principle.

Here is your super short intro to the work-energy principle. First, let’s look at work. Work is a way to add or take away energy from a system. The work depends on both the magnitude of the force and the direction the object is moving. Let’s say that the human travels a distance *d* during this catch. In that case, the gravitational force will do positive work (since it is pulling in the same direction as the displacement) and the firefighter will do negative work (pushing up in the opposite direction as the motion).

But what about the energy? For this system (of just the falling human), there is only one type of energy—kinetic energy. The kinetic energy depends on both the mass and the speed of the faller. The idea is to have the total work done on the human decrease the kinetic energy to zero (so that the human stops). Now to put it all together, it looks like this (yes, I skip a bunch of details).

I already have the estimated speed (from above) so I just need the human mass and stopping distance. Let’s say this is a human that isn’t super big—maybe 50 kilograms. For the stopping distance, it looks like the firefighter grabs the falling human and moves about 1.5 meters before coming to a stop. With these values, the force the firefighter needs to exert on the human would be 1,478 Newtons. For you imperials, that is about 330 pounds. It’s a large force, but not impossible. Still very impressive for just one hand.

Oh, and don’t forget that if the firefighter pulls on the human with almost 1,500 Newtons, the person pulls on the firefighter with the same force in the opposite direction. This means that the hero has to hold onto the window sill in order to not get pulled out of the building and fall along with the victim. Yes, there does appear to be a harness on the firefighter but it doesn’t look like it has tension. Still a superhero in my mind.

I have one final comment. Since I used the work-energy principle to estimate this force it seems like this is a good time to add an important note about energy. Remember—energy isn’t a real thing. It’s just something that we can calculate the can be conserved in many situations. There. I said it.

If you like science fiction, I can recommend a show for you—*The Expanse*. It takes place in the not-so-distant future all right here in our own solar system. There are no pew-pew lasers or faster-than-light space travel. When humans are on a spacecraft, they either “float” around or use magnetic boots (except when the spacecraft is accelerating). There are no “inertial dampeners” in *The Expanse*. Not only that, but it has interesting characters and a compelling plot. I like it.

As it turns out, *The Expanse* has three seasons all on the SyFy Network—but they did not renew for season 4. My plan was to write a physics piece about *The Expanse* to encourage another studio to pick it up. It seems my plan might have already worked—as Amazon Studios might be taking over. Hopefully.

OK, now for some physics. Let’s look at this flash back scene that shows the invention of the Epstein drive. The basic idea is that the space craft use some type of nuclear fusion rockets and this dude figured out a way to make them more “efficient”—I guess that means more thrust with less fuel. But why can’t the pilot move his hand during this acceleration?

Let me start with a seemingly completely unrelated experiment. Here are two cars on a low friction track. They are just sitting there. The track is level and they are not moving and not accelerating. Boring, but important.

I should point out that there are magnetic bumpers on these two cars. These magnets can push the cars apart when they get close—but right now they are far enough apart that there is no force. You can think of this magnetic bumper like a spring. In fact, I would have used a spring but I didn’t find a suitable one.

These two cars represent parts of a human’s body. There is no “compression” between these two body parts so the human would feel “weightless.” This human is in deep outer space far away from any large gravitational objects so that the human is indeed actually weightless.

What about a human standing on Earth? Here are the same two cars with the track inclined a bit. There is a large block that prevents the red car from moving (this would be like the floor on Earth).

There is really only one difference in this case in that the two cars are closer together. The “magnetic spring” has to be compressed some (which you can see with my paper-scale) in order for the red car to push “up” on the blue car. The human in this case would *not* feel weightless. The human would feel normal.

Hopefully it’s clear that I am trying to make a human feeling model. The distance between these two cars is a measure of how a human “feels”—at least in terms of weight.

Are you ready for the next case? What if I put these two cars on a level track and then push one of the cars with my finger so that it accelerates? Here’s what that looks like.

I’m pushing the blue car to the right so that it accelerates. But what about the red car? It also accelerates to the right, but I’m not pushing it. Instead the red car accelerates from this “magnetic spring” between the two cars.

This is what happens when you have human in a car (an actual car) that is speeding up. The seat pushes forward on the human and then the internal parts of the human push on each other. I’m sure you’ve been in an accelerating car before, right? You know what it feels like. It feels sort of like the car is leaning back. This acceleration feels exactly like gravity because both compress that spring between your body parts. And there you have Einstein’s Equivalence Principle: an accelerating reference frame is equivalent to a gravitational field.

And here is your answer to the crushing acceleration of the Epstein drive. The acceleration of the spacecraft is just like a super high gravitational field. On the surface of the Earth, the gravitational field pulls mass down 9.8 Newtons for every kilogram (9.8 N/kg) and we call this “1 g” since the gravitational field uses the symbol “g.” This would be equivalent to an acceleration of 9.8 m/s^{2}. So, if you feel 8g’s that would be the same as a planet in which you weigh eight times as much on Earth. That means your hand that normally has a weight of 5 Newtons would feel like it’s 40 Newtons (1 pound to 8 pounds).

Of course you have to lift more than your hand to turn off an accelerating spacecraft (especially when you disable the voice commands). The whole arm might have a normal weight of 35 Newtons (8 pounds) such that it would feel like 284 Newtons (64 pounds). While some people might be able to lift a 64 pound dumbbell, a man that was living on Mars probably couldn’t. The gravitational field on the surface of Mars is only 3.8 N/kg—you don’t have to be quite as strong to move around on Mars as you do on Earth.

But wait! I have one more case to point out how humans feel weight. Let’s go back to the two cars on the track. What would happen if I let them accelerate by rolling down an incline? Here’s what that would look like.

Here both cars are accelerating close to the same value as when I pushed one of them. However, the magnetic spring is *not* compressed. This situation represents a human in a gravitational field *without* a floor—such as a person in free fall or an astronaut in orbit. In both cases there is a gravitational force on the human but this gravitational force causes the human to accelerate. There is a big difference between acceleration due to gravity and an acceleration due to some other force. For the two cars, the gravitational force pulls on *both* cars to cause them to accelerate. There is no need for a compressed magnetic spring to make the other car (remember these cars represent body parts) to accelerate. Since there is no spring compression, you (the human) would feel weightless. And yes, this is why astronauts feel weightless in orbit even though there is indeed gravity in space.