Find second object's velocity in relativity

[Istiak]

Homework Statement

A particle of mass M at rest decays into two particles of masses m1 and m2 travelling in
opposite directions at velocity v1 and v2 respectively. Express v2 in terms of v1, m1, m2, and M.

Relevant Equations

$m=\frac{m_0}{\sqrt{1-(\frac{v}{c})^2}}$

> A particle of mass M at rest decays into two particles of masses m1 and m2 traveling in opposite directions at velocity v1 and v2 respectively. Express v2 in terms of v1, m1, m2, and M.

Since both objects are from a single object that's why I took relativistic mass of both objects are same. I was thinking if there's some other simple equation to solve this but I couldn't find it.

$\frac{m_2}{\sqrt{1-(\frac{v_2}{c})^2}}=\frac{m_1}{\sqrt{1-(\frac{v_1}{c})^2}}$
After rearranging the equation I get

$\frac{m_2}{m_1}=\frac{\sqrt{c^2-v_2^2}}{\sqrt{c^2-v_1^2}}$

But that's not the correct answer. So what's the correct one? What the concept should be to solve the question?

 

[Orodruin]

Gather all places in your mind where relativistic mass occurs and wipe them clean. It is a concept that is quite deprecated. If you need to use something similar, the total energy of the object will do. See https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

It is not the case that the energy of both particles need to be the same after the decay. Do you know any other conservation laws that may be applicable?

 

[Istiak]

Gather all places in your mind where relativistic mass occurs and wipe them clean. It is a concept that is quite deprecated. If you need to use something similar, the total energy of the object will do. See https://www.physicsforums.com/insights/what-is-relativistic-mass-and-why-it-is-not-used-much/

It is not the case that the energy of both particles need to be the same after the decay. Do you know any other conservation laws that may be applicable?

Thanks. Relativistic momentum was helpful. I got what I was looking for. And I knew that <a href="https://physics.codidact.com/posts/285324">Relativistic mass is outdated</a> but I couldn't think much that moment while solving that problem that's why I randomly took rel mass but at first sight I was looking for relativistic velocity, but that wasn't good idea tough.