In a pool game, the cue ball, which has an initial speed of 5.0 m/s, makes an el

Question

In a pool game, the cue ball, which has an initial speed of 5.0 m/s, makes an elastic collision with the eight ball. After the collision, the eight ball moves at an angle of 30o to the right of the original direction of the cue ball. Find the direction of motion of the cue ball and speed of each ball immediately after collision.

Explanation / Answer

A) Let the velocity of the 8 ball is v m/s and the velocity of the cue ball is u m/s at A* angle from the initial direction of the cue ball after the collision:- By the law of momentum conservation:- (i) In the initial direction(x-axis):- =>m x 5 = m x v x cos30* + m x u x cosA* =>5 = 0.87v + ucosA* =>ucosA* = 5 - 0.87v ------------(i) (ii) In the other direction(y-axis):- =>vsin30* = usinA* =>usinA* = 0.5v ------------------(ii) By the law of energy conservation:- [as the collision is elastic] =>1/2 x m x (5)^2 = 1/2 x m x u^2 + 1/2 x m x v^2 =>25 = u^2 + v^2 --------------------(iii) By (i)^2 + (ii)^2:- =>u^2 = [ 5 - 0.87v ]^2 + (0.5v)^2 {by putting the value of u^2 from (iii)} =>25 - v^2 = 25 + 0.75v^2 -8.7v + 0.25v^2 =>2v^2 -8.7v = 0 =>v(2v-8.7) = 0 =>v = 4.33 m/s [as v=/=0 ] [answer for B] By putting the value in (iii) =>u = sqrt[6.25] =>u = 2.5 m/s [answer for B] By putting the value of u & v in (ii):- =>2.5sinA* = 4.33/2 =>sinA* = 0.866 = sin60* =>A* is 60* {opposite side of 30*} [answer for A]

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