Have I not made it clear how much I like the MythBusters? Also, I am totally aware that they are not (nor do they claim to be) scientists. Really, this is what makes their show appealing (maybe?). So here is the problem now. And, it is not just the MythBusters – I see other shows making the same mistake.
If two things are colliding, how do you characterize the collision? No one really gets this right. For this particular episode, the MythBusters were looking at the collision between a bullet and gun. They wanted to see how hard it would be to shoot a gun out of someone’s hand. The question was: what kind of impact would it take to knock the gun out of the hand? No one wants to actually hold a gun while someone shoots it, so they tried to do something comparable. And here is the problem. What has to be the same for it to be a comparable collision. This is what Jamie came up with: a baseball bat. And here is his analysis:
Let us ignore the error confusing potential and kinetic energy. Clearly, he means kinetic energy. Well, I understand what he is trying to say. He is saying this baseball bat has the potential (when it hits) to create the following collision. But, this is a bad idea. The kinetic energy of an object does not characterize what kind of collision it will have. Suppose I cover the baseball bat with really squishy foam. Further suppose that this doesn’t significantly change the mass (or I shave off some of the wood to keep the mass constant). Also, assume that I could swing it the same speed, then it would have the same kinetic energy. However, this squishy bat collision would be significantly different than a hard bat collision.
So, kinetic energy of an object is not a good method for characterizing a collision. But before I go on. Let me check Jamie’s math – you know, just because. He says the bat has a mass of 800 grams and a speed of 85 mph. Kinetic energy can be calculated as:
I would like to calculate this kinetic energy in Joules. To do that, I need the mass in kg and the speed in m/s. I have a more detailed post on unit conversions, but google calculator does an awesome job also. The mass of the bat is 0.8 kg, and the speed is:
This gives a kinetic energy for the bat = 578 Joules. Again, I can use google calculator to check this in foot-pounds and I get 426 foot-pounds. Ok, good enough (even though that is a weird unit for energy)
Doing the same for the bullet, I get a kinetic energy of 563 Joules. Close enough.
What about momentum?
When people think about collisions, momentum often comes up. Would this be a good way to characterize a collision? Again, no. The same reason as above. If I have a baseball bat covered with foam, I could make it have the same momentum as the bullet (at a different speed) but it would have a different effect during the collision. How fast would the bat have to be moving to have the same momentum (note that I am pretending like it is a free object and not being held by the bat swinger).
And now for the bat:
I know what you are thinking. If an object had the same momentum AND kinetic energy as the bullet, it would have the same collision. Again, this doesn’t work. Think of two objects with the same mass and same speed. These would have the same momentum and same kinetic energy. Now what if one of these is a pillow and one is a brick? Will the interact the same with a gun that they hit? No.
The answer for collisions
I think the best thing to characterize these collisions is momentum of the object and time of impact. For the case of the gun in the hand, really you are asking “how much force would I need to exert on the gun to make it fall out”. This comes straight from the momentum principle. If I look at the forces acting on the thrown object (say it is a bat), then:
If I assume the only force acting on the bullet is the target object, then you can see that for the same change in momentums, you can have very different forces (depending on the time). Let me make an extreme example. Suppose I can push a car and run really fast (I can push a car, but I can’t run very fast). And suppose I push this car for 2 minutes while exerting a net force of 300 Newtons (I think I could do that). If the car started from rest, it would have a momentum of: (I am going to drop the vector notation and just deal with the x-direction)
Now suppose you are standing up against a wall and the car crashes into and takes 1 second to stop. Here is a picture:
I guess I should have pushed the car the other way so that my positive x-axis would be to the right. Oh well. Now, what force would be exerted on the person if the car stops in 1 second?
This is the same mistake Jamie makes. He assumes that if you shoot a bullet, the forces from the shooting are the same as when the bullet collides with the gun. So, he made a special gun that shoots from the side to simulate getting hit by a bullet. This is his first version that actually failed to fire (but has the concept).