A body of mass M moving at a speed V₀ collides elastically with a mass m at rest. After the collision, the two masses move at angles θ₁ and θ₂ with respect to the initial direction of motion of the body of mass M. The largest possible value of the ratio M/m, for which the angles θ₁ and θ₂ will be e…

Question

A body of mass M moving at a speed V₀ collides elastically with a mass m at rest. After the collision, the two masses move at angles θ₁ and θ₂ with respect to the initial direction of motion of the body of mass M. The largest possible value of the ratio M/m, for which the angles θ₁ and θ₂ will be equal, is

Accepted Answer

(1) 3. Perpendicular momentum with equal angles: Mv₁ sin θ=mv₂ sin θ⇒ Mv₁=mv₂. Along the motion: MV₀=(Mv₁+mv₂)cos θ=2Mv₁ cos θ⇒ v₁=(V₀)/(2 cos θ), v₂=(MV₀)/(2m cos θ). Kinetic energy cannot increase: 1/2Mv₁²+1/2mv₂²≤1/2MV₀²⇒1/(4 cos²θ)(1+/Mm)≤1. Since cos²θ≤1: 1+/Mm≤4, so /Mm≤3.

Step-by-step solution

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