If a sphere of mass `m` moving with velocity `u` collides with another identical sphere at rest on a frictionless surface, then which of the following is correct ?

Answer

77

Explanation

In `DeltaAQR, sin 30^@ = (QR)/(10//3)`
`QR = 5//3 u = 50 m//s ("Given")`
At the highest point P, the velocity of the bullet = `u cos theta`
Applying conservation of linear momentum at the highest point
`M(u cos theta) + 3M xx 0 = (M + 3M)v implies v = (M u cos theta)/(4M) = (u cos theta)/(4)`
At R, T = 0 but `v != 0`
`:. 4 Mg cos 60^@ = (4 MV^2)/(l) implies V^2 = (gl)/2 = 50/3`
Applying energy conservation principle for P and R
`Delta K = Delta U = 0`
`implies 1/2 (4 M) v_R^2 - 1/2 (4M )v_P^2 + (4M)gh = 0`
`=1/2 (4M) (u^2 cos^2 theta)/(16) = (4M) g xx (10/3 + 5/3) + 1/2 (4 M) (50/3)`
`implies cos^2 theta = 0.75 " "implies " "cos theta = 0.86 implies theta = 30^@`
`R/2 = (u^2 sin 2theta)/(2g) = (50 xx 50 xx sqrt(3))/(2 xx 10 xx 2) = 108.25 m " " H = (u^2 sin^2 theta)/(2g) = (50 xx 50 xx 1)/(2 xx 10 xx 4) = 31.25 m`.