A proton of mass m and initial velocity v0 collides with a helium atom of mass 4

Question

A proton of mass m and initial velocity v0 collides with a helium atom of mass 4m initially at rest.  After collision, the proton leaves the point of impact at an angle of 30 degrees with the original line of motion, find the final velocities of the proton and helium atom.  Assuming that the collision is perfectly elastic.  What will be the velocities of the proton and helium atom when the collision is inelastic and the energy loss is equal to one fifth of the initial energy of the proton?  Scattering angle of the proton remains at 30 degrees

Explanation / Answer

ELastic Collision

(a) Conserving the momentum in horizontal direction :

mp x vo = mp vcos 30 +mHe Vx

vo = vp cos 30 + 4 Vx

SImilarly in vertical direction

0 = vp sin 30 + 4 Vy

vp = -8 Vy

Conserving KE

1/2 m vo ^2 = 1/2 m vp^2 + 1/2 (4m) (Vx^2 + Vy^2)

ie vo^2 = vp^2 + 4(Vx^2 + Vy^2)

(b) Incase of inelastic collision

We can use same momentum equations

For KE

we have

(1/2 m vo ^2) x 0.8 = 1/2 m vp^2 + 1/2 (4m) (Vx^2 + Vy^2)

ie .8 vo^2 = vp^2 + 4(Vx^2 + Vy^2)

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