Relativistic Momentum of a particle

[~Sam~]

Homework Statement

A particle of unknown mass M decays into two particles of known masses m1 = 0.5 GeV/c2 and m2 = 1.0 GeV/c2, whose momenta are measured to be p1 = 2.0 GeV/c along the positive y-axis and p2 = 1.5 GeV/c along the positive x-axis. Find the unknown mass M and its speed.

Homework Equations

p=1/sqrt(1-v^2/c^2)(mu)

The Attempt at a Solution

I tried summing the momentas, via pythagoras theorem. after that I'm not sure.

 

[zhermes]

What's the general principle which governs how you can find the answer? I.e. for a given mass 'M,' why wouldn't m1 = 500000 GeV and m2 = 100000 Gev?

Hint: what if you think about the decay reaction as a collision?

 

[~Sam~]

Ahh I got it...using conservation of energy.

 

[ashishsinghal]

use conservation of momentum and energy to get the velocity in ground frame

 

[rwisz]

I can provide a response to this question by using the principle of conservation of momentum and the equation for relativistic momentum. The total momentum of the system before and after the decay must be equal. Therefore, we can write the equation:

p1 + p2 = p

Where p is the momentum of the unknown particle M. We can also write the momentum of the two particles in terms of their masses and velocities:

p1 = m1v1
p2 = m2v2

Substituting these equations into the first one, we get:

m1v1 + m2v2 = p

Using the equation for relativistic momentum, we can write:

p = 1/sqrt(1-v^2/c^2)(m1u1 + m2u2)

Where v is the velocity of the unknown particle M and u1 and u2 are the velocities of the two particles after the decay. Since we know the values of m1, m2, p1, and p2, we can solve for u1 and u2:

u1 = p1/m1 = 2.0 GeV/c / 0.5 GeV/c^2 = 4 c
u2 = p2/m2 = 1.5 GeV/c / 1.0 GeV/c^2 = 1.5 c

Substituting these values into the equation for p, we get:

p = 1/sqrt(1-v^2/c^2)(0.5*4c + 1.0*1.5c) = 1.5/sqrt(1-v^2/c^2)c

Now, we can solve for v by equating this expression to the total momentum of the system:

p1 + p2 = p = 1.5/sqrt(1-v^2/c^2)c

Solving for v, we get:

v = 0.8c

Finally, we can use the equation for relativistic energy to solve for the mass of the unknown particle M:

E = mc^2 = sqrt(p^2c^2 + m^2c^4)

Substituting the values of p and v, we get:

M = sqrt((1.5/sqrt(1-0.8^2)c)^2 + m^2c^4)/c^2

Solving for M