# Explosion of a moving mass into two parts.

I figure this problem will not be easily solvable which is why I'll ask.

Basically a moving mass M with a velocity along the x direction of .5c explodes into two fragments. Fragment A moves to the -x and fragment B moves to the +x. The sum of the mass is 1/4A + 3/4B = M. What are the final velocities of the two fragments?

I made this question up. Most physics books tell you the Mass is at rest initially but what would happen if it moves and then explodes. What are the calculations for it?

mathman
You haven't supplied enough information - need energy (kinetic energy of fragments) after explosion. Once you have that you can carry out the calculation assuming it started at rest. Then add the initial velocity to both pieces, making sure you take into account special relativity if the fragments are moving fast.

need energy (kinetic energy of fragments) after explosion.
But that would require the velocity as well. Couldn't the system be solved in the rest frame and then moved over into the moving frame. I'm assuming that in the rest frame the velocities of the two fragments can be found along with the relativistic KE and then moving the system over into a moving frame of reference is just a relativistic velocity addition?

I just wanted to point out that the explosion does not add any momentum to the two fragments and that the system is conserved. Lets just say that the two parts move apart instead of "explode".

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mathman
What makes them move apart? Although the total momentum in the c.m. coordinate system is zero, how much each piece has depends on the energy resulting from the explosion.
If there is no energy, they won't move apart at all.

What makes them move apart? Although the total momentum in the c.m. coordinate system is zero, how much each piece has depends on the energy resulting from the explosion.
If there is no energy, they won't move apart at all.
I understand, its because of the first reason you explained. I did make the problem up so its not something thats from a book, but thanks for the help.

PAllen
One issue is your problem statement is incomplete and confusing. A possible guess (but I shouldn't have to guess, and this may not be what you meant) is:

I have an object with rest mass M. It splits by some unspecified process into objects A and B, whose rest masses are 1/4M and 3/4M, respectively.

If this is what you meant (but didn't say - read your first post), the problem is already determined. In the rest frame of M, there is no available energy given the masses of the pieces. So A and B remain where they are, just separated by imagination. This is they way it will look in any other frame relative to which M is moving: an imaginary crack has appeared in M.

To allow A and B to move in the COM frame, you must specify rest masses for them whose total is less than M.

Maybe you meant something different, but I couldn't make head or tail of what that might be.

[Note, you could make this well defined and have something closer to what your intent might be if you specified e.g. that:

rest mass of A= .99*.25 M, rest mass of B=.99*.75M. The problem would then be fully determined given all your other constraints.]

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One issue is your problem statement is incomplete and confusing. A possible guess (but I shouldn't have to guess, and this may not be what you meant) is:

I have an object with rest mass M. It splits by some unspecified process into objects A and B, whose rest masses are 1/4M and 3/4M, respectively.

If this is what you meant (but didn't say - read your first post), the problem is already determined. In the rest frame of M, there is no available energy given the masses of the pieces. So A and B remain where they are, just separated by imagination. This is they way it will look in any other frame relative to which M is moving: an imaginary crack has appeared in M.

To allow A and B to move in the COM frame, you must specify rest masses for them whose total is less than M.

Maybe you meant something different, but I couldn't make head or tail of what that might be.

[Note, you could make this well defined and have something closer to what your intent might be if you specified e.g. that:

rest mass of A= .99*.25 M, rest mass of B=.99*.75M. The problem would then be fully determined given all your other constraints.]

No you're right, I just assumed the two masses moved apart with 1/4M and 3/4M which wouldn't work since something has to make that happen. So obviously if the system is conserved then (1/4M and 3/4M) < M and to make it conserved some extra energy must be added to the system to make the two parts move away from each other.

PAllen
No you're right, I just assumed the two masses moved apart with 1/4M and 3/4M which wouldn't work since something has to make that happen. So obviously if the system is conserved then (1/4M and 3/4M) < M and to make it conserved some extra energy must be added to the system to make the two parts move away from each other.
Just to outline how it is fully specified as long as you give some rest mass to A and B such that: A + B < M [letting A and B stand for the given rest masses].

Then you have, by energy conservation (using frame where M is initially at rest):

Mc^2 = E1 + E2 [E1 and E2 are total energy for A and B respectively: KE + rest energy]

and by momentum conservation:

E1^2/c^4 - A^2 = E2^2/c^4 - B^2

These two equations determine E1 and E2 in terms of A, B, and M. Then you have momentum (absolute value) of A and B (same, different direction) by:

p^2 = E1^2/c^2 - A^2 c^2

Then v1 = -p/√(A^2 + p^2/c^2)
v2 = p/√(B^2+p^2/c^2)

Then using velocity addition, you get the velocities in your chosen frame where M is moving at .5c.

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I'll even give you that:

E1 = (c^2/2M)( M^2 - (B^2 - A^2))

E2 = Mc^2 - E1

Then just plug in whatever numbers you want.

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