Solving Collisions & ZMFs Homework: Mass 4 Particle Deflection

[villiami]

Homework Statement

A particle of relative mass 4 travels at velocity v, and collides with a stationary particle with relative mass 1. By considering the zero momentum frame, show that the larger particle can be deflected by an angle of arcsin(1/4) at the most.

(Note this is a non-relativistic problem)

Homework Equations

The Attempt at a Solution

 

[rl.bhat]

The problem can be solved by using conservation energy and momentum.
Lat v1i is the initial velocity of mass 4 and v2i is the initial velocity of mass 1. You can wright three equations.
4v1i + 0 = 4v1f*cos(theta1) + v2f*cos(theta2) [ moments along the x direction]...(1)
4v1f*sin(theta1) = v2f*sin(theta2)[y-components of momentum]...(2)
0.5*4*v1i^2 = 0.5*4*v1f^2 + 0.5*v2f^2 [conservation of energy}...(3)
Now rewright equation(1) as 4v1i - 4v1f*cos(theta1) = v2f*cos(theta2) and square it. Now square equation (2) and add it to the above equation.After simplification you will get
5v1f^2 -8v1i*v1f*cos(theta1) + 3v1i^2 = 0. Here v1i is constant. For real root of this quadritic equation we must have [64v1i^2cos^2(theta) -4*5*3v1i^] > 0
[[64v1i^2{1 - sin^2(theta)} -60v1i^] > 0 Taking 4v1i^2 common we get
16 - 16sin^2(theta) - 15 > 0 or 1 - 16sin^2(theta) > 0 or sin(theta) < 1/4. That is the required result.