Elastic collision of neon atom problem

[mithanon]

Homework Statement

A neon atom (m = 20.0 u) makes a perfectly elastic collision with another atom at rest. After the impact, the neon atom travels away at a 57.9° angle from its original direction and the unknown atom travels away at a -45.4° angle. What is the mass (in u) of the unknown atom? [Hint: You can use the law of sines.]

variables
m = neon = 20 u
M = mass of unknown
$$\theta$$ = 57.9°
$$\phi$$ = 45.4°
v = initial velocity of neon
v' = final velocity of neon
V = final velocity of unknown

Homework Equations

I've set up conservation of momentum equations for both directions, and the energy equation, but I'm stuck with more unknowns than equations

The Attempt at a Solution

mv²=mv'²+MV²
mv = mv'cos$$\theta$$ + MVcos$$\phi$$
0 = mv'sin$$\theta$$ - MVsin$$\phi$$

using a momentum vector sum diagram and law of sines? I got
mv/sin(180-$$\theta$$-$$\phi$$)=mv'/sin($$\phi$$)

not sure how that helps but I guess I can then express v' in terms of v, eliminating one variable

no idea where to go from here

 

[hage567]

Everything looks OK so far I think. You can also use the law of sines to get the other "side" of the triangle. The problem is now just one of algebra. I would pick one equation (maybe the energy one), and substitute in for the variables you don't want and then isolate M.

 

[thomate1]

, help!

I would suggest using the conservation of energy equation to solve for the mass of the unknown atom. Since the collision is perfectly elastic, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision. This would give us an additional equation to use in solving for the mass of the unknown atom.

We can also use the law of cosines to relate the velocities of the neon atom before and after the collision. This will give us another equation to use in solving for the mass of the unknown atom.

Once we have these two equations, we can solve for the unknown mass by setting them equal to each other and solving for M. This will give us the mass of the unknown atom in terms of the mass of the neon atom and the angles of deflection.

In summary, to solve this problem, we can use the conservation of energy equation and the law of cosines to relate the velocities and angles of deflection of the neon and unknown atoms. This will allow us to solve for the unknown mass and complete the problem.