# Momentum of massless particle after decay

• nonequilibrium
In summary, a particle of mass M decays into two particles, one with mass m and the other massless. The momentum of the massless particle is found to be the opposite of the momentum of the particle with mass m. Using conservation of energy, it is determined that the momentum of the massless particle is equal to (M^2 - m^2)/2M.
nonequilibrium

## Homework Statement

I must be overlooking something very simple:
"A particle of mass M decays from rest into two particles. One particle has mass m and the other particle is massless. The momentum of the massless particle is..."

## Homework Equations

energy² = mass² c^4 + p² c²

momentum is conserved

## The Attempt at a Solution

a quick drawing makes clear that $\vec p_{m}$ and $\vec p_{\gamma}$ (with gamma denoting the massless particle) are on the same line, and since the initial momentum was zero, we get $\vec p_{m} = - \vec p_{\gamma}$ and thus $p_m^2 = p_\gamma^2$.

Using conservation of energy we get $M^2 c^4 = m^2 c^4 + p_m^2 c^2 + p_\gamma^2 c^2 = m^2 c^4 + 2 p^2 c^2$ with p the sought for momentum.

However, the answer should be $p = \frac{M^2 - m^2}{2M} c$

mr. vodka said:
Using conservation of energy we get $M^2 c^4 = m^2 c^4 + p_m^2 c^2 + p_\gamma^2 c^2$

I don't think that this is what conservation of energy gives.

You have put the squares of the energies added together. Conservation of energy is the energies themselves added together. You will then see that when you square it, you get cross terms. In other words, what is correct is:

$$E_i=E_{1,f}+E_{2,f}$$

What you have written there is:
$$E_i^2=E_{1,f}^2+E_{2,f}^2$$

You are missing the cross term.

Oh jeesh, thank you

Things are simpler if the the conservation of energy equation is rearranged before squaring.

## What is momentum?

Momentum is a physical quantity that measures the motion of an object. It is the product of an object's mass and velocity.

## What is a massless particle?

A massless particle is an object that has no rest mass. This means that the particle is always moving at the speed of light and has no mass at rest.

## What happens to the momentum of a massless particle after decay?

After decay, the momentum of a massless particle remains the same. This is because massless particles always travel at the speed of light, so their momentum cannot change.

## How is the momentum of a massless particle calculated?

The momentum of a massless particle is calculated using the formula p = E/c, where p is momentum, E is energy, and c is the speed of light.

## What are some examples of massless particles?

Some examples of massless particles include photons, which are particles of light, and gluons, which are particles that hold together the nucleus of an atom.

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