I always keep an eye on upcoming movies with interesting physics. In this case, it’s Captain America: Civil War. The most recent trailer is fine, but let me look back at an earlier clip that shows Captain America jump-kicking some dude on top of a car. Yes, there is physics here.
Momentum and Collisions
Even though Captain America is kicking someone, the interaction can be considered a collision. We can represent this interaction as a force. Since forces are an interaction between two objects, the force that Cap pushes on the dude (that’s his official name now) is the same force that the dude pushes back on Captain America.
If the collision is short, then this kick force is the only significant force (gravity does pull on both, but the time interval is short). Looking at just Captain America, I can use the Momentum Principle in the x-direction to say:
Here I am using px2 as his momentum after the kick and px1 as the momentum before. Assuming that his mass doesn’t change during this interaction, I can write the momentum in terms of his velocity.
So here’s the plan. If I measure the velocity of Captain America before and after the kick, I can find his change in momentum. Since the dude has the same force for the same time, he will have the same (but opposite) change in momentum—if there are no external forces. Let’s just see what happens.
Oh, but what about the energy from the kick? This just means that Captain America will add energy to the system so that the total kinetic energy after the kick could be greater than the kinetic energy before the kick. It’s like an explosion—but momentum is still conserved.
I need to get velocity values from the video. The best way to do this is with video analysis. My preferred tool is the free Tracker Video Analysis software. The basic idea is to examine the location of an object in each frame of the video to get position and time data. There is one small problem with this video—the camera moves. This means I must account for the motion of the object with respect to the background. Fortunately, Tracker Video Analysis has a method for dealing with these situations using calibration points. All I need to do is to scale the video (based on the size of Captain America’s shield), then move the coordinate system in each frame.
There are two objects to consider—Captain America and the dude. Let’s start with a plot of Captain America’s vertical position as he jumps onto the truck.
By fitting a quadratic function to this data, I can get the vertical acceleration. The term in front of t2 would be 1/2 the acceleration. This gives a vertical acceleration of about 12 m/s2—which is higher than the expected value of 9.8 m/s2. I’m not upset. For a video like this, that’s not too bad. Also, maybe my scale for the shield is off. Let’s move on.
What about the horizontal speed of Cap before and after the kick? Here’s what I get.
Oh, constant velocity. That’s cool, right? The slope of this line is 4.09 m/s (9.15 mph)—so that seems reasonable. But at what point did Captain America collide with the dude? It’s not at the end, it’s at the location with the small arrow. Yes, his speed didn’t even change. Was that because he kicked the dude? No. If he kicked the dude, then the dude would have pushed back on him (in the negative x-direction) and this should decrease his x-momentum. But it didn’t. OK, let’s look at the motion of the dude.
From this, the dude has a recoil speed of 8.15 m/s (18 mph). The recoil speed isn’t a problem, it’s the total momentum. Suppose that the dude and Captain America have the same mass (it’s close enough to be true). In this case, I can write the following expression for the x-momentum both before and after the kick.
Really, the only way Cap’s velocity can be close to the same before and after the kick is if the dude has no mass (or if Cap’s mass is much much larger than the dudes). So clearly there is a problem here. I will fix this by determining the correct final speed of Captain America. Using the equation above, I can solve for vc2 (the c is for Cap) and I get:
Captain America’s kick would push him back in the opposite direction. He wouldn’t just keep moving in the same direction.
Modeling a Kick
You don’t really understand something until you make a model. Here is a python model of Captain America kicking the dude. I set the mass of the both objects to 75 kg, but this is something you should change and play around with. The model starts with Captain America moving through the air. When he gets close enough to the dude, there is a force pushing them apart. Oh, there isn’t a gravitational force on the dude just to make it look like the clip. (Just click the play button to see the code run.)
You can see that if momentum is conserved, Captain America should recoil back after the kick. The only way to fix this would be to have some external force pushing him forward during this kick—maybe like a rocket pushing him. OK, there is another explanation. Maybe the dude has a wire that pulls him back when Captain America kicks him. Yes, I know this is only a movie and not real life. That doesn’t mean I shouldn’t analyze it.