It’s another year and another Star Wars Day—May the 4th be with you. Following my tradition, I’m going to take some element from Star Wars and do some cool physics. For this year’s post, I’m going to look at the end of The Empire Strikes Back. The great thing about using this movie is that it’s so old—more than 40 years—that I don’t have to worry about spoilers. I mean, if you haven’t seen it by now, are you really going to watch it?
So, here is the scene: Leia, Lando, and Chewbacca use the Millennium Falcon to escape from the Imperial forces on Bespin. On their way out, they grab Luke (he was literally just hanging around). Once they get off the planet, of course, Darth Vader is there to intercept them with his Star Destroyer. Lando says, “Oh, no biggie. We will just make the jump to lightspeed and skip out of this system.” Well, that doesn’t work. The Imperials have disabled the hyperdrive.
R2-D2 is the real hero here. He’s onboard the Falcon talking to the Bespin central computer—you know, just sharing lubrication techniques and dropping some gossip on the silly things C-3PO says. The central computer comes back with a rumor: The hyperdrive has been turned off. So now R2 knows what to do. He rolls over, and with the flick of a switch—boom. There goes the Falcon, right off into hyperspace. Hopefully they’re looking where they’re going and won’t hit a planet or something.
Now for the cool physics. When the starship makes the jump to hyperspace, R2 goes flying backwards inside the Falcon. It’s as though he was on a turbocharged bus when the driver hit the gas, and he’s not seatbelted in. If we take the inside of the bus as the reference frame, then we will need to add a fake force to account for the acceleration. I mean, it’s not necessarily a fake force. According to Einstein’s equivalence principle, there’s no difference between an accelerating reference frame and a gravitational force.
So, in the reference frame of the accelerating Falcon, there appears to be a gravitational-like force that pushes in the opposite direction as the acceleration. The magnitude of this force on R2 would be equal to his mass multiplied by the acceleration of the spaceship. If R2 has completely frictionless wheels (or at least very low friction), then as the Falcon accelerates forward he would accelerate backwards with respect to the ship’s frame. That’s a good thing—because I just need to measure R2’s acceleration as seen from inside the spacecraft.
This means we get to do some video analysis. If I know the size of stuff inside the Falcon, then I can determine the position of R2 in each video frame. Also, with a known frame rate I can get the time for each of these positions. For the distance scale, I’m going to use the height of R2-D2 and the frame rate that is embedded in the video (so that it plays back at the correct speed). My favorite tool for getting this data is Tracker Video Analysis. (It’s free.) Of course, there are some small issues with this analysis. The camera pans and zooms—but I can compensate for that motion by looking at how R2 moves with respect to the wall. With that, I get the following plot of position vs. time:
Notice that the data looks like a parabola? Yes, if an object has a constant acceleration then the position-time data should be parabolic. By looking at the coefficient in front of the t2 term, I can determine the acceleration. In this case, R2 is moving back with an acceleration of about 4.78 meters per second squared. That means that the Falcon should be accelerating in the forward direction with this same value—but of course it’s not. There’s no way it could make the jump to hyperspace with such a small acceleration. Obviously, there are some type of “inertial dampeners” that compensate for the motion of the spacecraft. Otherwise, every jump to hyperspace would kill everyone inside as they all got flung to the back of the vehicle. (Yes, I borrowed the inertial dampeners term from Star Trek. I don’t think they ever actually mention them in Star Wars.)
But wait! There’s another way to look at the acceleration in the interior of the Falcon. When R2 turns on the hyperdrive, we also see Leia and Luke in the cockpit getting thrown back. In fact, the camera even shows the floor tilting back. Yes, since this fake force due to the acceleration is pushing backwards, it’s essentially the same thing as an inclined plane. Let me show you how we can calculate the effective tilt angle of this ship as it makes the jump to hyperspace.
There are actually two fake forces inside the ship. First, there is the backwards-pushing force that’s parallel to the ground—this is due to the forward acceleration of the starship. The second fake force is the one that pushes down towards the floor of the Falcon. I guess this is due to some type of artificial gravity that is beyond my understanding (but very useful when filming a movie that takes place in space). These two forces can be combined into a single, net fake force that is angled downwards and backwards. Here, I made a diagram to show you these forces:
The angle this net force makes with respect to the floor is very similar to the components of a real gravitational force down an inclined plane. In fact, if you were R2-D2, it would seem just like this spaceship is stationary and tilted up at an angle. Like this:
I am going to assume that the floor-pointing artificial gravity (F1) is just like on Earth, with a magnitude of mass of the object (R2-D2) multiplied by the gravitational field of g = 9.8 m/s2. The horizontal force (F2) will be the mass of R2 multiplied by the acceleration of the Millennium Falcon. From these two forces, both of which I know, I can calculate the effective tilt angle. With my value of the Falcon acceleration, I get an incline of 26 degrees. Cool, right?
Now let’s go back to Leia and Luke in the cockpit. In the scene, you can see the interior of the Falcon tilting back. Based on my very careful measurements, it seems like the floor leans back 6 degrees, assuming the camera doesn’t tilt. This doesn’t match the acceleration of R2-D2—and that’s OK, I’m just trying to figure out how they filmed this. Did they actually tilt the studio-based Millennium Falcon cockpit or did they have the actors throw up their hands and fall back while tilting the camera in the opposite direction? Did they tilt the floor and let the R2-D2 prop roll downhill? Did they pull it with a string? I don’t know. If I had to guess—and, apparently, I do—I would say they tilted the floor for R2 and let it accelerate downhill because it’s the easiest way to get a constant acceleration. However, the incline angle was much lower than my calculated value of 26 degrees. So to make it look cooler, they just played back the film at a higher speed to make it look faster. Either way, it’s still a great scene.
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