As I’ve probably hinted with my previous post on baseball physics [1], I love the show “Mythbusters”. This past week, we had a new 2-hour episode focused on car demolition myths. In one segment, Adam and Jamie revisit an old myth that two semis, heading at one another at 50 mph and striking a compact car at the same time, can fuse metal to metal and make the compact car seem to disappear. They showed that the myth itself was not possible, despite perfect timing on the semi collision. They then tried to find out how much force would make the myth possible. What if the two trucks were driving at the speed of sound?
To do this, they put the car at one end of a track and fired a two-stage rocket sled at the car, achieving 650 mph before the sled hit the car. Their goal: to simulate the energy of two trucks smashing together at the speed of sound, with compact car stuck in the middle.
The same thought process that applied to the baseball myth applied here. The energy of the two trucks colliding is like two particles colliding head on with the same energy – the same velocity. Since kinetic energy is given by 1/2mv*v, where m is the mass of the object and v is the velocity of the object, the goal is to do the simulation with a fixed car and moving sled that simulates the energy of the two semis colliding head-on on the stationary car.
The energy of a head-on collision of two objects of equal mass is given by E = E1 + E2, where E1 and E2 are the kinetic energy of the two objects (trucks in this case). If the collision occurs in our laboratory as one object moving while the other is stationary, one needs to convert that energy into the “effective” energy of the same collision happening has a head-on collision. To get the same energy from the case where one object is at rest and the other is moving, you have to bring the kinetic energy of the moving object WAY up. Did the Mythbusters achieve this?
Let’s compute the ratio of the two cases: E_fixed/E_colliding = (1/2 M V*V)/(1/2 M v*v + 1/2 M v*v) = 1/2 V*V/v*v. This assumes the rocket sled had the same mass as one of the semis. We want the ratio to be 1, which indicates the simulation reproduces the target collision of a speed-of-sound collision.
Let’s run with that assumption. If the trucks were moving at the speed of sound and colliding head on, their speed would be v = 760mph, while the rocket sled was moving at V=650 mph. That gives us an energy ratio of these two cases of 0.4. That’s lower than they wanted. To really nail this myth, they would have needed to get the speed up to about 1000 mph on the rocket, assuming a sled with the same mass as a semi.
Let’s take the ratio of the masses of the semi and rocket sled into account. A semi weighs many tons – according to one estimate [2], an empty semi weighs about 7 tons. The rocket sled probably weighed no more than a few tons – lets say 2.
Our ratio then becomes 1/2 (M/2m) (V*V/v*v) = 0.05.
More way off. Eh. This is, of course, all math fun with kinetic energy. I love that the Mythbusters go from colliding beams to fixed target experiments, involving tremendous forces when they do so. While I might argue with their simulation of the colliding beam phenomenon, it sure is fun to watch a compact car vaporize when a rocket sled hits it at 650 mph.
Update (2009/04/14): this is what I get for blogging later at night. The equation for energy ratio in the case where the sled is not the same mass as a semi should have read:
(1/2 M V*V)/(1/2 m v*v + 1/2 m v*v) = (1/2) (M/m) (V*V/v*v)
I was wrong by a factor of 1/2 in the original equation (stupid factors of 2!). Thus, the energy ratio of the case they wanted to test to the case they did test was 0.1, not 0.05. So they were only off by a factor of 10, not 20.
