Question

A lump of putty with a mass of 0.370 kg and a speed of 0.870c collides head-on and sticks to an identical lump of putty moving with the same speed. After the collision the system is at rest. What is the mass of the system after the collision?

Answer 1

Conservation of (mass + energy).

Call the (initial) mass of each lump,
m = 0.370 kg

In the center-of-momentum (CM) frame, where both bodies have initial speed,
v = 0.870c,
the total initial mass-energy, in mass terms, for each body is
γm = m/√[1-(v/c)²]

The total initial mass-energy, in mass terms, for both bodies, then, is
2γm = 2m/√[1-(v/c)²]

So assuming there is no other form of energy that escapes the site of the collision, the final mass of the lump, which will be at rest in the CM frame, by conservation of momentum, is equal to that:

M[final] = 2m/√[1-(v/c)²] ≈ 1.5 kg

Now this all seems paradoxical, because the lumps still have all the same atoms and molecules, which all have the same masses. So where could the extra mass come from?

The answer is that the collision resulted in an incredibly huge production of heat energy (in fact, the lumps would be well beyond vaporized here!), and although the same 740 g of atoms are present, no more, no less; they are moving with tremendous kinetic energies, so that the entire collection is a body whose heat energy content is
(1500- 740) g·c²
but whose overall momentum is 0 in the frame we're using.

So it responds physically as a single body of mass =1500 g.