The epic fight over the tropical planet of Scarif in Rogue One remains one of my favorite battles in all of Star Wars. Since there is a small chance you’ve not yet seen the movie, I will give you a spoiler alert.
Still here? Excellent. Let me explain the scene. Two star destroyers wage a pitched battle against a whole bunch of rebel ships. Several Y-wing starfighters disable one of the star destroyers, and a hammerhead corvette shoves it into the other one. It’s a great move by the rebels, but I’m not here to discuss military tactics. I’m here to talk about physics, and this battle offers a great opportunity for that.
I will break it down into three parts: the collision, the debris trail, and the motion of the star destroyer.
The rebels in the hammerhead clearly braced for impact. The Imperial officers did not, and they careen across the bridge. It looks cool, but what would really happen?
First, let’s assume the star destroyer has a significantly greater mass than the hammerhead. I’d guess it is on the order of 100 times more massive. And mass matters. When the hammerhead strikes the star destroyer, it exerts a force on the destroyer. Because force changes the momentum of an object (where momentum is the product of mass and velocity), the star destroyer moves in the same direction as the hammerhead.
Forces are an interaction between two objects. This means that when the hammerhead pushes on the star destroyer, the star destroyer pushes back with the same magnitude force and thus the same change in momentum. But the same change in momentum does not mean the same change in velocity. Since the hammerhead (probably) has a smaller mass, it will experience a much greater change in velocity than the star destroyer. This means the impact should toss the hammerhead’s crew around even more than it tosses the Imperial guys around.
Maybe something else is going on. Perhaps the ships use an inertia field of some kind to prevent everyone from experiencing high accelerations. Clearly, something keeps people on the floor, because there appears to be gravity aboard the ships. All of which is to say I still love this scene even if the physics don’t work out.
If you want some homework, I think you could estimate the recoil speed of the star destroyer by looking at one of the Imperial dudes flying across the bridge. It should be fun.
As the hammerhead pushes the star destroyer, random stuff falls away from the star destroyer. Although visually appealing, would it actually happen?
If both ships are orbiting Scarif (which probably is not the case, given that they stay near the shield gate, but just go with it for the sake of argument), it would be just as if they are in a region with no gravitational forces. Because there is no air in space, there would be no drag forces acting on the debris either—so they should move at a constant speed and remain near the star destroyer instead of falling away. Ah, but the star destroyer is not moving at a constant speed. It is accelerating due to the force of the hammerhead.
Can you actually see accelerating debris? Let’s use video analysis to examine the pixel-position of one object in each frame. Using the time for each frame, I can determine how the debris moves relative to the star destroyer. If I approximate the scale of the star destroyer and I label the direction directly away from it the negative y-direction, then I get the following plot for a piece of junk that comes off the ship:
I expected a straight line indicating a constant (and not physically correct) motion. However, the plot shows a parabola. Excellent. The position vs. time graph for an object with a constant acceleration should indeed be a parabola. I can even use the coefficients of the fitting equation to estimate the acceleration of the star destroyer with a value of 5.34 m/s 2. But wait! The debris is also moving and accelerating in the x-direction with a value of 2.2 m/s 2 (of course, these values depend upon my estimation of the scale). Combining these two components of acceleration, I get a total magnitude of 5.78 m/s 2 (you must use the Pythagorean theorem). That’s a pretty high acceleration, but it is indeed accelerating.
More homework: Use this value of acceleration to estimate the thrust force from the hammerhead’s engines.
The star destroyer is of course mostly rigid. So when you push on it with some force, you can assume no other significant forces act on it. With that in mind, what happens? It accelerates, yes, but it also should experience a change in rotational motion.
When you take an introductory physics course, you typically start with simple stuff. When you push on an object, you assume that object is merely a point in space with no dimensions. This works surprisingly well in a whole lot of situations—like balls, cars, sliding blocks, and other stuff. In these cases, you can use the momentum principle. But the hammerhead and the star destroyer are not point mass objects. In this case, you use the angular momentum principle. It states that a torque on an object changes that object’s angular momentum.
OK, what is torque? Consider it a rotational force. The torque depends on both the force that is pushing and the location where that force is applied. But you already knew this, even if you don’t realize it. You know that to open a door, you push on the side farthest from the hinge. By increasing the distance from the hinge (this distance is called the torque arm), you increase the torque. It would be silly to try and open the door by pushing on the hinge. Even with a huge force, you have a tiny torque. Yes, I have done this before. So have you. Admit it.
What about angular momentum? This is a lot like linear momentum (which physicists just call momentum). It is a product of the moment of inertia and the angular velocity—which is a measure of how fast something is rotating. The moment of inertia tells you how difficult it is to change its rotational motion. I like to call it the rotational mass.
How do you model an object that is both accelerating and changing rotational motion? You can make a numerical model. In this case, I will estimate the force from the hammerhead and find the change in momentum and angular momentum after a brief time interval (say 0.01 seconds). After that, I can approximate the position and angle for the star destroyer. After that, I repeat the process a whole bunch of times until I get the motion of the starship. Yes, I find it best to do this with a computer.
Here is my model. Click the play button to run it and the pencil to look at the code.
If you want to play with the code, go ahead. I included some notes suggesting things you might change. But overall it looks like it works. I like to say that you don’t really understand something until you can model it. Oh, but notice that the tip of the star destroyer isn’t just moving in a circle. This is because the star destroyer is simultaneously rotating about the center of mass and moving its center of mass.
But how does this model compare to the movie? It’s difficult to measure the motion of the entire rotation, but I can at least plot the angular position of the star destroyer right before the collision.
This shows the star destroyer rotating with a fairly constant angular velocity at about 0.27 radians per second. This doesn’t quite agree with my model for two reasons. First, in my model the angular velocity is not constant but keeps increasing. Second, my value for the final angular velocity was 0.096 radians per second. I could adjust the thrust force from the hammerhead, but I will leave to you.
Oh, one more thing. Notice in my model above, the hammerhead provides a thrust force of 2 x 10 11 Newtons. For comparison, the Saturn V rocket exerts a thrust of 3.3 x 10 7 Newtons. Just to be clear, the hammerhead creates about 6,000 times the force of the Saturn V.
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