Angle of particles after elastic collision

[ajl1989]

Homework Statement

A particle of mass M1 collides elastically with a particle of mass M2 at rest. Show that if M1>M2 then the angle of M1 after the collision cannot exceed the value arcsin(M2/M1).

Homework Equations

conservation of momentum: M1v₁=M1v₁`+M2v<sub>2</sub>`
conservation of kinetic energy: (1/2)M1v₁²=(1/2)M1v₁`<sup>2</sup>+(1/2)M2v<sub>2</sub>`²

The Attempt at a Solution

I've figured out that the velocities after collision are:
v₁`=v<sub>1</sub>*[(M1-M2)/(M1+M2)]
and
v<sub>2</sub>`=v₁*[(2M1)/(M1+M2)]
but I don't know how to find the maximum angle of M1 after collision. Please help

 

[Pythagorean]

Homework Statement

A particle of mass M1 collides elastically with a particle of mass M2 at rest. Show that if M1>M2 then the angle of M1 after the collision cannot exceed the value arcsin(M2/M1).

Homework Equations

conservation of momentum: M1v₁=M1v₁`+M2v<sub>2</sub>`
conservation of kinetic energy: (1/2)M1v₁²=(1/2)M1v₁`<sup>2</sup>+(1/2)M2v<sub>2</sub>`²

The Attempt at a Solution

I've figured out that the velocities after collision are:
v₁`=v<sub>1</sub>*[(M1-M2)/(M1+M2)]
and
v<sub>2</sub>`=v₁*[(2M1)/(M1+M2)]
but I don't know how to find the maximum angle of M1 after collision. Please help

what's the critical point between when M1 > M2 is true, and when it's no longer true?

what will M1 - M2 be at that point?

to get angle into your equation, think projection.