A particle decays into two daughter particles.Find velocity pf daughter particles?

humanist rho

1. The problem statement, all variables and given/known data

A particle of rest mass m moving with speed c/2 decays into two particles of rest mass 2m/5 each.The daughter particles move in the same line as the direction of motion of the original particle.Then what are the velocities of daughter particle?


2. Relevant equations

Momentum conservation:

p = p₁+p₂

Energy conservation:
(p²c²+m²c⁴)^1/2=(p₁²c²+m₁²c⁴)^1/2+(p₂²c²+m₂²c⁴)^1/2


3. The attempt at a solution

Can i assume that both daughter particle has same momentum(velocity)?
otherwise the second equation becomes really complicated.:(

vela

Try solving the problem in the rest frame of the original particle, and then transform back to the lab frame.

humanist rho

In the frame of original particle, the daughter particles will have equal and opposite momentum.

Then?how to proceed?

vela

What are the energy and the magnitude of the momentum of each particle?

humanist rho

velocity of the particles in the frame of original particle,[tex]v=\frac{\frac{c%
}{2}+v^{\prime }}{1+\frac{v^{\prime }c}{2c^{2}}}[/tex]
magnitude of momentum,[itex]p=\frac{2}{5}\frac{mv}{\sqrt{1-\frac{v^{2}}{c^{2}}}}[/itex]
Energy =[tex]\left( p^{2}c^{2}+m^{2}c^{4}\right) ^{1/2}[/tex]

am i correct?
But things seems much more complicated.

vela

You're kind of going backwards. First, what is the energy of the particle in its rest frame? Then by conservation of energy, what can you say about the energy of the decay products? Then calculate their momenta.

humanist rho

At rest frame,

Energy = mc²

By conservation of energy,

[tex]2(p^{2}c^{2}+\frac{4}{25}m^{2}c^{4})^{1/2}=mc^{2}[/tex]

[tex]p=\frac{3}{10}mc[/tex]

is that right?

humanist rho

and velocity of the daughter particle in this frame,
[tex]v=\frac{3}{5}c[/tex]

humanist rho

And on converting it to lab frame,i got,

[tex]v=\frac{c}{7}[/tex]

I think now i started to understand. Thanks a lot

vela

Looks good. You'll find in general that when trying to solve problems in relativity like this one, it's best to stick with energy E and momentum p when you can and only resort to working with velocity v/c when absolutely necessary.

To be a bit pedantic, this is the speed of each daughter particle. You still have to account for their directions.

What's the speed of the other daughter particle in the lab frame?

humanist rho

In the frame of original particle,
Each new particle has velocity ,±5c/3 .

In the lab frame one particle has velocity +11c/13 and other has velocity -c/7.

Why aren't they equal? even when both of them have same mass and is produced from same particle?

niklaus

If they were equal in the lab frame that would violate conservation of momentum, since the original particle has non-zero momentum...

humanist rho

yes i understood. thanks. :)