Finding loss of energy in collision

1. Aug 29, 2013

Saitama

1. The problem statement, all variables and given/known data
A particle of mass $m_1$ experienced a perfectly elastic collision with a stationary particle of mass $m_2$. What fraction of the kinetic energy does the striking particle lose, if it recoils at right angles to its original motion direction.

(Ans: $2m_1/(m_1+m_2)$ )

2. Relevant equations

3. The attempt at a solution
Let the initial velocity of $m_1$ be $v$ and let the x-axis be along the initial direction of motion. After collision, the first particle flies off at right angles and let that direction be y-axis. The vertical component of velocity of $m_2$ after collision has the direction opposite to that of $m_1$. Conserving momentum in x direction:
$$m_1v=m_2v_{2x}$$
Conserving momentum in y direction:
$$m_1v_1=m_2v_{2y}$$
where $v_1$ is the final velocity of $m_1$. I still need one more equation.

Any help is appreciated. Thanks!

2. Aug 29, 2013

Curious3141

What quantity, apart from momentum, is conserved in a perfectly elastic collision?

3. Aug 29, 2013

Saitama

Energy. But that gives a very dirty equation. I have seen some problems on one dimensional collisions where coefficient of restitution is used to find another equation. Is it possible to apply the same here as I think it reduces the algebra work greatly.

4. Aug 29, 2013

Curious3141

Energy, as in total energy, is *always* conserved.

In a perfectly elastic collision, kinetic energy is specifically conserved.

The algebra is fairly easy to work out here. Took me less than 10 lines and barely 5 minutes.

Remember that the final speed of $m_2$ is given by the Pythagorean theorem. Deal only in squares of the velocity components, and everything simplifies quickly.

And always keep in mind what you're trying to find, which is the ratio $\displaystyle \frac{v^2 - v_1^2}{v^2}$.

5. Aug 29, 2013

Saitama

I have solved the problem using the energy approach, thanks a lot Curious!

6. Aug 29, 2013

Curious3141

You're welcome, and I'm glad you solved it. Sorry I couldn't stay up to continue to help, but I need early nights as I've not been in the best of health lately.

7. Aug 30, 2013

Saitama

I hope you get well soon. :)