It’s May 4, so happy Star Wars Day—may the fourth be with you!
One of the iconic scenes from Star Wars: Return of the Jedi is the battle on Tatooine at the Sarlacc Pit, the home of a massive creature that just waits to eat the things that fall into its sand hole. (No spoiler alert: It's been almost 30 years since Return of the Jedi hit the theaters. If you haven't seen it by now, you probably aren't going to.)
Luke Skywalker is being held captive by Jabba the Hutt’s guards. They’re on a skiff above the Sarlacc Pit, and Luke is standing on a plank, about to be pushed into the creature’s maw. R2-D2 is some distance away on Jabba’s sail barge—and he has been keeping Luke's lightsaber. Now for the best part: At just the right moment, R2 launches Luke's lightsaber so that it flies across the pit for Luke to catch. As that happens, Luke jumps off the plank and spins around. He catches the edge of the plank and uses it to springboard himself into a flip back onto the skiff. Now the battle begins.
I'm going to look at these two motions—the lightsaber toss and the plank flip—and see if it's possible for an ordinary human to do this, or if you have to be a Jedi like Luke. But I am going to make one big assumption about this scene, and you might not like it. I'm going to assume that the planet Tatooine has the same surface gravity as Earth, so that g = 9.8 newtons per kilogram. This would mean that a jumping human and a thrown lightsaber would follow similar trajectories on both planets.
Oh, I get it: Tatooine is not the same as Earth. However, in the movie it looks a lot like Earth (you know why), and this allows me to make some actual calculations. Let's do it.
I'm going to start with the lightsaber that R2-D2 launches towards Luke. What can we figure from this part of the action sequence? Well, let's start with some data.
First I'm going to get the total flight time as the lightsaber moves from R2 to Luke. The simplest way to do this is to use a video analysis program; my favorite is Tracker. With this, I can mark the video frame that shows the weapon leaving R2-D2's head (which is kind of weird when you think about it) and then mark the frame where it gets to Luke. This gives a flight time of 3.84 seconds.
I'm going to assume that's not the actual flight time. Why? First, it's a pretty long time for the lightsaber to be in the air. Also, there's quite a bit happening during that shot. In the sequence seen in the movie, R2-D2 shoots the saber and we see it rising. Cut to Luke doing a front flip onto the skiff. Cut to Luke landing, then a shot of the lightsaber falling towards him. The final shot shows Luke's hand catching the weapon. That's a lot of cuts, and so it might not be a real-time sequence. Don't worry, that's fine. That's what movie directors do.
But there's another way to look at the motion of the lightsaber. If I know the size of R2-D2 (which I do—he is 61.7 centimeters wide), then I can use that to find the position of the lightsaber in the video frames while it is in the air. With that, I get the following data:
Since this is a plot of the vertical position (y) as a function of time (t), the slope of this line would be the vertical velocity. That puts it at 8.11 meters per second. (Rebels don't use Imperial units, but just in case you do, that's 18.14 miles per hour.) That's about the speed of a ball tossed by an ordinary human.
With this vertical velocity, we are almost ready to figure out how long the lightsaber should be in the air. But we need one more assumption. Since R2 is on top of Jabba's sail barge and Luke is on a skiff floating below it, the lightsaber will need to land some distance below its starting height. I'm going to approximate a change in height of 3 meters, which seems plausible. Now I can use the following kinematic equation for objects with a constant acceleration, like a free-falling lightsaber:
In this equation, y1 is the starting position and y2 is the final position. Let's set the final position to 0 meters so that the starting position would be 3 meters. The initial velocity (vy1) is going to be the value of 8.11 meters per second, and g is the gravitational field (9.8 N/kg = 9.8 meters per second2). The only thing I don't know is the time (t).
It takes a little bit of work to solve this, using the quadratic equation. Doing so gives a flight time of 1.10 seconds. Notice that this is indeed a shorter time interval than the value from the clip (3.84 seconds). I think this interval is more legitimate.
Now we can look at the horizontal motion of the lightsaber. In this case, the lightsaber is a simple projectile. Since there are no forces acting on it in the horizontal direction, it travels with a constant horizontal velocity. That means that if we know the horizontal distance between Luke and R2, we can calculate the horizontal velocity just by dividing this distance by the flight time (1.10 seconds). Let's say that it's 10 meters from the sail barge to the skiff. This would give the lightsaber a horizontal velocity of 9.09 m/s.
Knowing both the horizontal and vertical velocity at the launch, we can find the launch angle of the lightsaber. (This is something that R2 would have to calculate.)
Plugging in the numbers, this gives a launch angle of 41.7 degrees above the horizontal. That seems like a pretty reasonable shot—but it still feels like R2 launches it at a higher angle (like 70 degrees) to give Luke more time to get into position.
(Let's be honest: When they made this scene, they likely broke the lightsaber motion into two parts. The first shot shows the launch of the lightsaber as it went up into the air and then just landed somewhere. The second part was probably filmed as someone dropped the lightsaber into Luke's hand.)
Now let's move on to Luke's maneuver. We can also break this into two parts. In the first one, Luke steps off the plank while turning around. He starts to fall, then grabs the edge of the plank when he's at arms length below it. He uses the springiness in the board, along with his own muscles, to launch himself to an even higher position. In the second part of the move, he does a front flip back onto the skiff so he can be in position to catch his lightsaber.
Let’s focus on that plank-grab move. I can illustrate this motion at three different points—start, grab, flip.
To make things as simple as possible, let's represent Luke as a point mass, with the location of that point somewhere above his belt line. So, in position 1, I will set this initial position as 0 meters. Once he drops, he gets down to a new position (y2) below this initial value. And finally he flips up to the highest point at y3.
There's a lot going on, but let's consider the simplest case by assuming a perfectly elastic plank that acts like a trampoline. In that case, it doesn't matter how far you fall. The plank just springs you right back to your starting position.
So Luke steps off the plank and falls, speeding up as he travels downward. He grabs the plank with his hands, and the force deforms it, causing it to act like a spring. This both stops his motion and stores elastic energy in the board. Then the plank pushes him upwards and converts the stored elastic energy into kinetic energy. This makes Luke move upward until he returns to his starting position, back at y = 0 meters.
But that's not going to be high enough for Luke to complete his awesome Jedi flip. He's going to need to get higher, up to position y3, if he wants to look cool in front of all these bad guys. That means he's going to have to add some energy from his own body into the system. The amount of energy (E) he will need to use is equal to the change in gravitational potential energy (Ug) going from position 1 to position 3.
(This also is exactly what non-Jedi humans do when they jump.)
We just need some estimates to calculate the change in energy. How about a mass of m = 70 kilograms, a gravitational field of g = 9.8 newtons/kilogram, and change in height (y3 – y1) of 0.5 meters?
The change in height is tricky. I think 0.5 meters might be enough to do a flip, but if you wanted to do a spectacular one, Luke might need to get a change in height of 1 meter. Let's go with the low end.
Putting these values in gives a change in energy of 343 joules. In real life, if you pick up a textbook off the floor and put it on the table, that takes about 10 joules of energy. Climbing one flight of stairs can be a change in energy of over 2,000 joules. So a 343-joule change in energy itself is not very impressive.
The difficult part is using that much energy in a short amount of time. We define the rate of energy as the power (in watts) where P = ΔE/Δt. So, we need to estimate the time that Luke is in contact with the plank and pulling on it to add enough energy to complete that flip.
Going back to the video analysis, getting this pulling time is pretty straightforward. It looks like Luke is actively pulling on the plank for 0.166 seconds. Now I can calculate the power he exerts during this pull:
Over 2,000 watts might seem like a large value. And in a certain sense, it is indeed high. Your coffee machine probably uses close to 1,000 watts when you are making your morning drink, and a hair dryer on high power uses about 2,000 watts. Regular humans produce an average of about 100 to 200 watts while exercising over a long period, like on a bike ride, but we can output 500 to 1,000 watts for very short intervals. So 2,000 watts isn't completely unbelievable. But what is impressive is that Luke is not using his strongest muscles—his legs. He's doing this with his arms.
And there’s one more thing: In the above calculation, I assumed the plank was perfectly elastic. It's clearly not. When Luke pulls down on the board, some energy is stored as elastic potential energy—but some of the energy also goes into other forms like sound, thermal energy, and general deformations of the material. As a rough approximation, we can assume that half of the energy from Luke's fall goes into actual elastic energy. That means that Luke will have to add in even more energy to make up for this loss.
If I assume he falls 2 meters before hitting the plank, this means that it will only push him back up 1 meter, because half the energy would be lost. Now he has to supply the rest of the energy to go from 1 meter below his starting point to 0.5 meters above that position for a total change in height of 1.5 meters. This would require an energy expenditure of 1,029 joules and a power of 6,199 watts. Now that is a power that no mere mortal could produce. Luke would have to draw strength from the Force. And that means that this move can’t be done by an ordinary human; you have to be an actual Jedi.
