How to Calculate Relativistic Momentum Using M, m, and c?

[Saptarshi Sarkar]

Homework Statement

A particle X of mass M at rest disintegrates into a particle of mass m and another massless particle. Calculate the momentum of m.

Relevant Equations

What I figured:

From conservation of mass-energy
$Mc^2 = \frac {mc^2} {\sqrt {1 - \frac {v^2} {c^2}}}+ pc$

From conservation of momentum
$0 = \frac {mv} {\sqrt {1 - \frac {v^2} {c^2}}} + p$

The answer is required to be in terms of M,m and c only. But, I am not able to calculate the momentum of the m mass particle using the above two. Can anyone help me by telling me what I am missing?

 

[PeroK]

Homework Statement:: A particle X of mass M at rest disintegrates into a particle of mass m and another massless particle. Calculate the momentum of m.
Homework Equations:: What I figured:

From conservation of mass-energy
$Mc^2 = \frac {mc^2} {\sqrt {1 - \frac {v^2} {c^2}}}+ pc$

From conservation of momentum
$0 = \frac {mv} {\sqrt {1 - \frac {v^2} {c^2}}} + p$

The answer is required to be in terms of M,m and c only. But, I am not able to calculate the momentum of the m mass particle using the above two. Can anyone help me by telling me what I am missing?

Try working with the quantities $E$ and $p$ for all particles involved.

 

[DEvens]

The answer is required to be in terms of M,m and c only. But, I am not able to calculate the momentum of the m mass particle using the above two. Can anyone help me by telling me what I am missing?

You are missing that you *can* calculate the momentum of m using those equations. Two equations, two unknowns.

 

[Saptarshi Sarkar]

You are missing that you *can* calculate the momentum of m using those equations. Two equations, two unknowns.

From the second equation I found the momentum to be equal to -p, but when I try to get the value to -p using the equation for conservation of mass-energy, I am not able to eliminate the v.

 

[PeroK]

From the second equation I found the momentum to be equal to -p, but when I try to get the value to -p using the equation for conservation of mass-energy, I am not able to eliminate the v.

Then don't have the $v$ in your equations in the first place. $v$ and $\gamma$ just get in the way.

 

[PeroK]

Can anyone help me by telling me what I am missing?

Probably the most important equation in SR is $$E^2 = p^2c^2 + m^2c^4$$
Which holds for both massive and massless particles.

 

[Abhishek11235]

For a more systematic approach,try working with conservation of 4 momentum and using the fact that(c=1):

$$P.P=m^2 $$
$$P= P_1 + P_2$$