I know it's just a movie, and not even a live-action one—but the trailer for Lightyear compels me to analyze it. This is an animated film about Buzz Lightyear. No, not the toy from Toy Story. This is about the real Buzz Lightyear that the toy is based on. (OK, I don't even know what's real anymore.)
But I do know that in the trailer for the movie, which is going to be released next summer, they show Buzz launching in his spacecraft, presumably from Earth. Since the "camera" view is far away, you can see a good bit of the rocket’s motion. This makes it a perfect case for video analysis.
The main idea behind video analysis is to look at the position of an object in each frame of a video. If I know the size of an object in the scene, I can scale the video to get an actual position of the object, or its x and y values. Then, after advancing to the next frame, I can find the object’s new position. Since the video changes frames at regular intervals, 24 frames per second, each new frame is 1/24 of a second after the previous one. That means I can get both x and y positions as a function of time from the video. It's kind of awesome.
But why should I get the position of Buzz's rocket as a function of time? I don't know what I expect to find, and that's what makes it so exciting. So let's get started.
I like to use Tracker Video Analysis. The first thing I need to do is to determine the video’s scale. I'm looking for an object near the spacecraft that is some known size. That's sort of difficult since everything in the scene is a computer animation—but it won't stop me. Let's use the spacecraft as our object of known size. In part of the trailer, you can see Buzz sitting in the cockpit. If I assume Buzz is about 1.8 meters tall (around 6 feet), then I can get a rough estimate that the length of the entire spacecraft is about 35 meters. That's good enough for now.
The trailer doesn't show a very clear view of the first part of the rocket’s launch, but soon after that, I can get some nice data. Here is a plot of the vertical position of the rocket as a function of time:
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This graph says that the vertical position of the rocket increases by a (nearly) constant amount from one frame to the next. In physics, we call that "constant velocity." Since this is a plot of position vs. time, the slope of the line will be equal to this constant vertical velocity. From the graph above, you can see this puts the launch speed of the rocket at 192 meters per second (m/s). That's pretty darn fast—but is that fast enough to actually reach space? The answer is both yes and no. Here’s why.
Let me give a brief overview of escape velocity. Suppose you take an apple and toss it up in the air with a velocity of 10 meters per second. (That’s fairly fast for an apple.) As that apple moves upward, it’s going to slow down. Eventually, thanks to the pull of gravity, it will stop and then start falling back toward Earth.
But let’s say the apple is moving super fast, at 11.186 kilometers per second. Then it will get high enough such that the gravitational force won’t be strong enough to stop it. That apple will escape.
Buzz Lightyear’s rocket is fast—but not that fast. Remember, we calculated that it’s moving at 192 meters per second. But that’s not a problem, because you don’t need to worry about escape velocity if you have a rocket. The engine will keep pushing the spaceship to overcome that pull and keep it moving at a constant speed, so it won’t fall back to Earth.
In the case of Buzz's rocket, there are essentially three force interactions during this part of the motion. First, there's the thrust from the engines. A conventional chemical engine combusts propellants to create exhaust gasses. All forces come in pairs, so when the exhaust is ejected from the engine, it pushes the rocket in the opposite direction. (The nice thing about rocket engines is that they work both in Earth's atmosphere and in space, where there is no air.)
The other two forces on the spacecraft are the downward-pulling gravitational force due to its interaction with the Earth, and an air resistance force pushing in the opposite direction as the ship. Air resistance is caused by the collisions between the rocket and the air.
As the spacecraft leaves the ground, both of these forces will eventually become insignificantly small. That’s because moving farther from the center of the Earth means that the strength of the gravitational force pulling on the ship decreases. And once the rocket gets beyond the atmosphere, there will no longer be air resistance, because there won’t be any air. The only force remaining will be the thrust from the engines, so the speed of the spaceship should increase.
But … this isn't how real rockets work. Normally, a rocket engine produces a thrust force that is greater than the gravitational force. This means that a rocket traveling upward would accelerate and not just travel at a constant velocity.
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Let's look at an example: the launch of the SpaceX Crew Dragon capsule atop a Falcon 9 rocket from May 2020. If I can analyze the motion of a fake movie rocket, I can also do video analysis for a real one. (All the details are here.) Since this SpaceX rocket has a fairly constant acceleration, I can create a plot of the vertical velocity as a function of time. The slope of this line would be the acceleration.
This gives the rocket an acceleration of 5.12 m/s2—that's fairly normal for real rockets.
But wait! The Buzz Lightyear rocket started from a resting state. Since it went from a velocity of 0 m/s to 192 m/s, that means it had to accelerate. Let's get a rough estimate of this acceleration. From the trailer, it looks like the spacecraft starts off at rest on the launch platform. After 2.5 seconds, it is off the platform and moving at its constant speed. Now we can use the following definition of acceleration:
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Putting in a change in velocity of 192 m/s and a time interval of 2.5 seconds gives an acceleration of 78 m/s2—which is a little bit more than the acceleration of the Falcon 9 rocket. What would that feel like? We can think about accelerations in terms of g-forces. An acceleration of 1 g is the equivalent of a human being stationary on the surface of the Earth (where g = 9.8 m/s2). You are probably at 1 g right now. If you were instead aboard the Crew Dragon as it was launched into space, you have an acceleration of 0.5 g's—but it would actually feel like 1.5 g's, because the Earth would still be pulling down on you until the rocket reached escape velocity.
Buzz Lightyear, on the other hand, would experience 8.9 g's. That’s huge, but it's survivable. Some fighter pilots can have maneuvers that pull up to 9 or 10 g's. (Plus, it's Buzz Lightyear, so he's probably tougher than your average fighter pilot.)
But now for the most important question: Why would the animators of Lightyear choose to create such an unrealistic launch? I mean, there are plenty of real-life launches that could be used as a basis for a cool animation, so it's not like they don't know what one should look like. I'll answer this question with another animation.
Here is a model I made in Python showing the Buzz Lightyear rocket and the SpaceX Falcon 9, both approximately to scale. The two rockets start from rest at the same time, but the Falcon 9 has a realistic acceleration and the Buzz Lightyear spacecraft has a motion based on the trailer. (If you want to look at the actual Python code, here it is.)
You see the Buzz rocket taking off and moving fast—like a rocket. On the other hand, the actual rocket doesn't look very impressive. Yes, sometimes real life just isn't good enough. So that's when the animators step in and ramp things up to make them look cool. Remember, the movie isn't a science lesson—it's a story. If the animators needed to change things up to make them look better, I'm all for that.
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