How to Derive Momentum Equations for Alpha Particles in Collisions?

[GayYoda]

Homework Statement

Fast moving α particles of mass m make collisions in a cloud chamber with gas atoms of mass M and negligible initial velocity. After a collision, the velocities of the scattered α particles and the recoiling gas atoms are v and V respectively, the former being inclined at an angle θ and the latter at an angle φ to the original α particle direction
show that 2vcos(θ+φ)=(1-M/m)V

Homework Equations

p=mv
cos(θ+φ)=cos(θ)cos(φ)-sin(θ)sin(φ)

The Attempt at a Solution

Resolving horizontally i got [1] ∑p = mvcos(θ)+MVcos(φ) and vertically i got [2] mvsin(θ)-MVsin(φ). i squared both equations and added them together to get m^2v^2+M^2V^2+2mvMVcos(θ+φ)=(Σp)^2. I'm not sure how to work out the initial velocity

 

[PeroK]

Is there anything in addition to conservation of momentum that you can use?

 

[GayYoda]

Energy of the system?

 

[PeroK]

Energy of the system?

Yes, conservation of energy.

 

[GayYoda]

how do i apply energy ?

 

[PeroK]

how do i apply energy ?

First, I think it would be simpler if you used a symbol for the initial speed of the $\alpha$ particle: $u$ seems a good choice to me.

Second, you must the formula for the kinetic energy of a particle.