You know I can't help but like Star Wars. Even with the new stuff, I watch it. Recently, I was watching the Clone Wars cartoon and noticed something odd about the way R2-D2 flies. I know what you are saying...."the odd thing is that he flies at all. Why didn't he fly in episodes 4-6?" Who knows. Here is the best image I could get of R2-D2 flying (from wookieepedia).
What is wrong? Well, maybe you can't tell from the image I posted. Here is a diagram of flying R2-D2.
If R2 (I can call him that because we are good friends) was flying like that, why would that be a problem? That would be a problem if he was going at a constant speed. I re watched the end of Clone Wars (the movie) and it seems like R2 is flying at a constant speed. Why is that problem? Let me draw a free body diagram.
If I assume that R2 is flying horizontally at a constant velocity, and the the "thrust" is at an angle theta from the vertical then:
I guess I should say I am using g as the local gravitational field for whatever planet he (R2) is on. Is R2 even a he? Maybe R2 is a she. I don't know. The point is that the vertical component of the thrust must equal the vertical component of the gravitational force. So what you say? Well, that leaves the horizontal component of the thrust so that the net force in the horizontal direction is:
Obviously I have missed the air resistance force on R2 while he is flying. That is what makes the net horizontal force zero, right? Well, let us calculate this. First, some assumptions (or maybe starting declarations):
- I would assume the world is like Earth because they seem to jump like a person on Earth. Needless, I will call the local gravitational field g.
- Also, I will assume other things are like the Earth. Most importantly, the density of air (I will call rho) - if it is even air.
- I am not sure what the air drag coefficient for R2 is, but I will call this C. I assume this value will be somewhere between a brick and a sphere.
- Let me call R2's linear speed v. This will need to be estimated later.
- I will need to estimate three things about R2, its mass ( m ) volume ( V ) and cross section area in the direction of motion ( A ).
- Finally, I need to know R2's thruster angle. I will call this angle theta from the vertical.
If R2 is moving at a constant velocity, then the forces in the x (horizontal) and y (vertical) directions will be zero. I will assume there is air resistance in the horizontal direction, and I will use the following model for the magnitude of air resistance:
So, the x- and y-direction equations for force are:
I will solve the y-equation for Fthrust and put that into the x-equation. Doing this, I get:
Let me tell you where I am going. If R2 is firing is thrusters a little forward and moving at a constant speed then I suspect R2 has a very low density and the thrusters are not pushing very hard. So, from the above stuff, I will solve for the mass.
Here are my estimations (you are welcome to come up with your own):
- rho = 1.2 kg/m3
- Area: Wookieepedia says that R2 is 0.96 meters tall. Using tracker video on an image of R2, I am going to approximate it as a rectangle that is 0.42 meters by 0.62 meters for an area of 0.26 m2
- Wikipedia lists the drag coefficient for a smooth sphere as 0.1. It has a smooth brick with a coefficient of 2.1. A skier has a coefficient of 1.0. Wikipedia does not list the drag coefficient for R2, but a value of around 1.0 seems reasonable.
- For the velocity, I took it a little far. I was just going to ballpark guess at his speed, but I didn't. I used Tracker to look at R2's motion in Clone Wars where he flies to rescue Padme. From this, I get a speed of 2.3 m/s.
- I already said I would assume Earth-like gravity. So, g will be 9.8 N/kg
- Theta is about 35 degrees (although it could be as high as 45 degrees).
Using these values, the mass of R2 is 0.1 kg. Yes, 100 grams. How do I know I am correct? I know because Wookieepedia doesn't list R2's mass or weight. They know it is silly, so they left it off.
If this mass is so low, I think R2 doesn't even need thrusters. He would just float (which would actually change my calculations above - I left off the buoyancy force). By my estimations, R2 is about .42 meters in diameter. This would put its volume at about 0.1 m3 and R2's density would be:
I was originally thinking that maybe R2 was made of styrofoam - but that has a density of about 40 kg/m3. So there.
I know someone is going to say "hey, chill man! It is just a movie. Don't ruin it by bringing in all your physics stuff." My reply, someone has already ruined the Star Wars movies, his name is George. Just kidding, I still like Star Wars.
There were two questions in the comments that I addressed in a second post. The first question: Was R2 flying at a constant speed - I included the data. Second: what was the angle of the thrusters.
