Check out this short clip from the next Captain America movie: The Winter Soldier from Yahoo.

At the end, Cap throws his shield at the Winter Soldier - because that's what Captain America does. But wait! The Winter Soldier just catches the shield and throws it right back at Captain America. The real cool part is what happens when Cap catches the shield. The impact is strong enough to push him back a little bit. Is this enough to get an estimate for the mass of the shield? I think so.

Sliding Back ————

This is really a multi-part problem. First, the shield is thrown by the Winter Soldier. I don't really care about the throwing motion. Next, the shield moves through the air to Captain America and collides with him. This gives him some recoil velocity. However, Cap is standing on the ground such that his recoiling body is slowed down to a stop by friction.

It might not seem to be the best place to start, but I am going to start backwards. Let's look at Captain America sliding after the impact with the shield. By estimating the frictional force and the sliding distance, I can get a value for the recoil speed after the impact.

In this first problem, I can just consider Captain America as a block with some initial speed moving across the ground. Here is a force diagram while he is slowing down (after the impact).

The forces in the vertical direction must add up to zero since Cap doesn't accelerate up or down. This means that I can find the force the ground pushes up on him:

Why do I need this force pushing up (usually called the Normal force)? If I use the typical model for sliding friction, the magnitude of the frictional force can be determined by:

The coefficient of friction (μ k ) is a value that depends on the two types of surfaces interacting. In this case, it would be between shoes and the ground (sounds like ground with gravel or sand on it).

Once I know this frictional force, I can use this in the x-direction to find the acceleration. I will say that Captain America starts the sliding with some velocity v 2 (in the negative x direction) and then ends at zero m/s (which I will call v 3 ). Why don't I use the starting velocity as just v 1 ? Don't forget, I am working backwards. We'll get to v 1 soon.

Now I can write the x-acceleration in terms of the forces:

Instead of deriving it, I'll just use one of the kinematic equations for motion in one dimension with a constant acceleration. Of course there is one that doesn't have a time variable in it, so I'll use that one.

When I put in the value for the acceleration in the x-direction, I added a negative sign since the acceleration is in the opposite direction to the velocity. But now I have an expression for the velocity of Captain America and the shield after the collision.

Shield Collision —————-

Now that I have an expression for the post collision speed, we can look at the collision. Here is a diagram showing Cap and the shield during the interaction.

Since this is an interaction between Cap and the shield, the two forces have the same magnitude but opposite direction. Further, the interaction time for the two objects is the same. This is important when considering the momentum principle. It says that (here I am ignoring the frictional force since it is over a short time interval).

With opposite forces and the same time, the two objects (Cap and the shield) must have opposite changes in momentum.

This is why momentum is conserved during a collision (assuming the external forces are small enough to ignore). Sometimes it is useful to rewrite this to say that the momentum before the collision is the same as the momentum after the collision.

Now back to the actual example of Captain America and the shield. Before they collide, Cap is stationary and the shield has some x-velocity of v 1 . After the collision, Cap and the shield have the same x-velocity of v 2 . In terms of momentum, I can write:

Just to be clear, I am using m s for the mass of the shield and m c for the mass of Captain America. Oh, and these are the horizontal components of velocity. Now, I can use this to solve for the mass of the shield.

Now I just need some values.

Estimations ———–

Before I look at the video, let me start with some basic estimations.

Captain America mass = 100 kg. (Last time, I guessed 90 kg, but I was corrected).

Coefficient of friction between Captain America and the ground = 0.4 (mostly guessing here).

Next, I need the speed of the shield as it is thrown at Captain America. I can use Tracker Video analysis to get a rough value for the speed. Wikipedia lists the diameter of the shield at 0.76 meters. If I scale the video with this value, I can get a few frame of motion. Here is a plot from Tracker that shows the horizontal position of the shield after being thrown by Winter Soldier.

This puts the initial shield speed at 19.5 m/s (43.6 mph). That's pretty fast for a shield (but we are talking about superheros here).

Now I need to estimate how far back Captain America slides after the collision. It turns out that the video clip does show Cap's foot as it's sliding. This is just enough to get an estimate of the sliding acceleration:

This puts the acceleration after the collision at around 3 m/s. With that, I don't even need to estimate the coefficient of friction (working backwards this would give a friction coefficient of 0.3). I can use the time of sliding and the acceleration. The time of sliding can easily be obtained from the video clip with a value of 1.08 seconds (wow, that seems fairly long). However, using this sliding time and the acceleration, I get the following for the post collision velocity of Cap:

I have values for everything except the mass of the shield. Plugging in all the stuff:

That's a pretty massive shield at around 43.9 pounds - but I guess that isn't too crazy for a metal shield. How about an estimate of the density of the material? Lets say the shield is a flat disk with a diameter 0.76 meters and a thickness around 0.5 - 1.0 cm (total guess here). This would put the volume of the material at 0.00227 - 0.00454 m3 giving a density range of 8767 - 4383 kg/m3. That seems ok. Iron has a density around 7874 kg/m3 and titanium is around 4500 kg/m3.

Still, that would be a heavy shield to carry around and even harder to throw.

Homework ——–

Here are some questions for you. Yes, these will all be covered on the final exam of SuperHero Physics 131.

Estimate the average force that Winter Soldier exerts on the shield while throwing it.

What is the approximate average force the shield pushes on Cap during the collision? You will need to estimate the collision time - you can get that from the video.

If Captain American can throw the shield at about the same speed as Winter Soldier (looks about the same), why doesn't the Winter Soldier recoil noticeably? Assume he recoils for just 0.25 seconds with the same acceleration as Captain America. Now estimate the Winter Soldier's mass.

Estimate the kinetic energy of a thrown shield. Estimate the power needed to make this throw.

If Captain America can throw a 19.9 kg shield at a speed of 15 m/s, how fast would he be able to throw a baseball? What about a football?

Redo my analysis assuming a coefficient of friction with a value of 0.4. What does this do to the mass of the shield?

That should keep you busy until Captain America: The Winter Soldier comes out in theaters.