Special Relativity GRE Question

[CreekDog]

Hi Guys,

I'm looking through one of the practice GRE tests, and I think that they have answered their own question incorrectly. I thought I'd get another opinion. Here it is:

36. A lump of clay whose rest mass is 4 kilograms is traveling at three-fifths the speed of light when it collides head-on with an identical lump going the opposite direction at the same speed. If the two lumps stick together and no energy is radiated away, what is the mass of the composite lump?
(A) 4 kg
(B) 6.4 kg
(C) 8 kg
(D) 10 kg
(E) 13.3 kg

I think it should be (C) since the lumps are presumably at rest after the collision, but the practice test says that the correct answer is (D). Doing the math for relativistic mass, a 4 kg lump would indeed have a mass of 5 kg when traveling 3/5c, but they're not moving after the collision. Am I missing something?

[Sorry, I'm new here. I just read the sticky, and see that this might not be the appropriate forum for my question. Rather than double post it, I'll wait to see if it gets moved -- apologies!]Thanks!

 

[nrqed]

Hi Guys,

I'm looking through one of the practice GRE tests, and I think that they have answered their own question incorrectly. I thought I'd get another opinion. Here it is:

36. A lump of clay whose rest mass is 4 kilograms is traveling at three-fifths the speed of light when it collides head-on with an identical lump going the opposite direction at the same speed. If the two lumps stick together and no energy is radiated away, what is the mass of the composite lump?
(A) 4 kg
(B) 6.4 kg
(C) 8 kg
(D) 10 kg
(E) 13.3 kg

I think it should be (C) since the lumps are presumably at rest after the collision, but the practice test says that the correct answer is (D). Doing the math for relativistic mass, a 4 kg lump would indeed have a mass of 5 kg when traveling 3/5c, but they're not moving after the collision. Am I missing something?

[Sorry, I'm new here. I just read the sticky, and see that this might not be the appropriate forum for my question. Rather than double post it, I'll wait to see if it gets moved -- apologies!]


Thanks!

It might be moved to the Homework forum.

In any case, their answer is right. The total energy must be conserved. The point i sthat some of the kinetic energy was transformed into rest mass energy. This is a key point of special relativity: mass is no longer conserved in general (as we are used to have in classical mechanics), but the total energy is conserved.

 

[CreekDog]

Ah, okay. Is this effect permanent, so long as the clay is stationary? Could you keep doing this to the lumps of clay, and keep making them more and more massive, even when they're at rest after the collision? I guess in the real world, they'd lose a lot of their kinetic energy to heat during the collision, though. Thanks for the reply.

 

[pervect]

Note that the answer given is correct assuming (as I think is highly likely) that the GRE was asking about the invariant mass of the two lumps of clay.

Recall the definition of invariant mass - in geometric units, it's just m^2 = E^2 - p^2. In non-geometric units that's (m c^2)^2 = E^2 - (pc)^2.The total momentum p of the system is zero. Ask yourself: what is the total energy, E, before the collision? If no energy is radiated away, what is the energy E after the collision?

 

[pervect]

Ah, okay. Is this effect permanent, so long as the clay is stationary? Could you keep doing this to the lumps of clay, and keep making them more and more massive, even when they're at rest after the collision? I guess in the real world, they'd lose a lot of their kinetic energy to heat during the collision, though. Thanks for the reply.

The "lumps of clay" in this problem will be relativistically hot. (That's where the energy in an inelastic collision goes - it turns into heat).

You are supposed to imagine, I guess, that an object that hot would still "hold together". Of course, the actual lumps of clay would turn into a ball of expanding plasma. If you were able to contain this plasma, though, and all the radiation it would emit in real life (recall, the problem specifies: no energy loss) the total mass would be 10 kg.

This is an example of how a hot object is more massive than a cold object, because of the energy you put into it to heat the object up adds to its energy, and hence to its mass (by which I mean invariant mass).

 

[CreekDog]

Thanks guys, I see now that I need to be thinking about the total energy before and after, and remembering that the total momentum is zero. Time to go review!