Question

A train of unknown initial velocity with a mass of 55000 kg slams into another non-moving train with an unknown mass. 27% of the kinetic energy of the moving train is lost. Both trains have wheels under them and friction is negligible (thus they are free-moving). Find the mass of the second, non-moving train.

Answer 1

There are two possibilities: train 1 bounces off train 2, and goes in opposite direction, or train 1 continues in same direction.

Suppose r is ratio of mass of train 2 to mass of train 1, ie train 2 mass is 55000r.

Let v0 be initial speed of train 1, v1 is final (post-collision) speed of train 1, v2 is final speed of train 2. (v0/v1)^2 = .73 since train 1 loses 27 percent of its kinetic energy, hence v1/v0 = .8775. Also, train 2 gets 27 percent of kinetic energy of train 1, so rv2^2=.27*v0^2, or r=.27*(v0/v2)^2. Those come from conservation of energy.

Now conservation of momentum. Putting sign on speed v1 to make it velocity,
m1*(v0 plus or minus v1)=m2*v2 which gives r=(.1225 or 1.8775)*(v0/v2).

Equating the two expression for r gives v0/v2 = .1225/.27 or 1.8775/.27,
that is v0/v2 = .4537 or 6.9537. Then r = .05558 or 13.05556,

Multiplying by 55,000 kg and adjusting to two sig figures, mass of train 2 is
either 3100 kg or 720000 kg. In the former case, train 1 continues along in the same direction, and in the latter case, train 1 bounces off train 2 and reverses direction.

That's my preliminary attempt. Still have to check results. In particular, want to do a reality check, to make sure that have not let the math produce an impossibility. In this situation, train 1 cannot end up with higher velocity than train 2, in the same direction; the math would not care, but the matter would object!

Answer 2

The right answer is 20342 kg. But it's OK I got it. I forgot to mention that its a completely inelastic collision :(

Answer 3

Thanks! Fun problem. Tiem for me to go to sleep!