# Momentum and Elastic Collisions

1. ### ndoc

13
1. The problem statement, all variables and given/known data
In a pool game, the cue ball, which has an initial speed of 8.0 m/s, make an elastic collision with the eight ball, which is initially at rest. After the collision, the eight ball moves at an angle of 30° to the original direction of the cue ball.

2. Relevant equations
V8 = Velocity of 8-ball
Vc = Velocity of cue ball

(1)Epx = m*V8*cos(30) + m*Vc*cos(x) = m*8
(2)Epy = m*V8*sin(30) + m*Vc*sin(x) = 0
(3).5*m*Vi^2 = .5*m*V8^2 + .5*m*Vc^2

3. The attempt at a solution
While these equations are technically solvable, they are nearly impossible by hand. Solving (3) for one velocity and using substitution twice I get:
sin(x)^2 -cos(30)*cos(x) + sin(30)*cos(30)*cos(x) - sin(30)*cos(x)^2 = -1/8

Is there an easier way to solve this since I know I will not be able to solve this equation myself?

2. ### rl.bhat

4,435
(1)Epx = m*V8*cos(30) + m*Vc*cos(x) = m*8
(2)Epy = m*V8*sin(30) + m*Vc*sin(x) = 0

Rewrite these two equations as
m*V8*cos(30) = m*8 - m*Vc*cos(x) -------(1)
m*V8*sin(30) = - m*Vc*sin(x) ----------(2)
Square both sides of eq.1 and 2 and add them. After simplification you will get the value of vc*cos(x)
From the conservation of energy equation, find the value of vc. Then you can find the angle x.

3. ### ndoc

13
Awesome, thanks so much!