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A particle of mass m moving with a speed u strikes a smooth horizontal surface at an angle (alpha).The particle rebounds at angle(beta) with a speed v.Determine the expression for v and beta if the coefficient of restitution is e.

A particle of mass m moving with a speed u strikes a smooth horizontal surface at an angle (alpha).The particle rebounds at angle(beta) with a speed v.Determine the expression for v and beta if the coefficient of restitution is e.

Grade:11

1 Answers

Harshit Singh
askIITians Faculty 5963 Points
3 years ago
Dear Student
Hereθindicates alpha andϕindicates beta.
Let velocity of particle in y direction after the collision bev′.
Initial velocity of the particle in x direction remains the same but its velocity in y direction changes.
∴ucosθ=vcosϕ ...............(1)
Also

(v′−0) / (−usinθ−0) = −e


⟹ v′=usinθ
∴eusinθ=vsinϕ ................(2) (∵v′=vsinϕ)
Dividing (1) from (2) we get
vsinϕ / vcosϕ​= eusinθ​ /ucosθ
⟹tanϕ=etanθ

Change in momentum of the particle
ΔP=m(vsinϕ)−m(−usinθ)
ΔP=m(eusinθ)+m(usinθ)=mu(1+e)sinθ
Impulse delivered I=ΔP=mu(1+e)sinθ
Squaring and adding (1) and (2), u^2cos^2θ+e^2u^2sinθ=v^2(cos^2ϕ+sin^2ϕ)

v^2=u^2(cos^2θ+e^2 sin^2θ)

This is the required expression.

Thanks

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