Help on physics collision problem finding degrees and speed given initial degrees and speed?

Question

In a pool game, the cue ball, which has an initial speed of 6.0 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 25° to the original direction of the cue ball.

(a) Find the direction of motion of the cue ball after the collision.

_____° (from the original line of motion)

(b) Find the speed of each ball. Assume that the balls have equal mass.

_____m/s (cue ball)

_____m/s (eight ball)

Answer 1

Apply the law of conservation of momentum along the original line of direction (assume this to be the x-axis):

M1V1 + M2V2 = M1V3(cos A) + M2V4(cos 25)

where

M1 = mass of the cue ball

V1 = initial velocity of the cue ball = 6 m/sec (given)

M2 = mass of the 8-ball

V2 = initial velocity of the 8-ball

V3 = final velocity of the cue ball

V4 = final velocity of the 8-ball

A = angle at which cue ball deviated from its original line of direction

Since M1 = M2

Since M1 = M2 then the above simplifies to

6 + 0 = V3(cos A) + V4(cos 25)

V3(cos A) + V4(cos 25) = 6 ---- call this Equation 1

Apply the law of conservation of momentum along the vertical of the original line of direction (assume this to be the y-axis):

0 = M1(V3)(sin A) + M2(V4)(sin 25)

Again, since M1 = M2, the above simplifies to

V3(sin A) = -(V4)(sin 25)  --- call this Equation 2

Since this is an elastic collision, then the kinetic energy of the system is constant throughout. Hence,

Initial kinetic energy = Final kinetic energy

(1/2)(M1)V1^2) + (1/2)(M2)(V2^2) = (1/2)(M1)V3^2) + (1/2)(M2)(V^2)

Since M1 = M2, the above simplifies to

6^2 = V3^2 + V4^2

V3^2 + V4^2 = 36 --- call this Equation 3

So, now you have three equations

V3(cos A) + V4(cos 25) = 6  ---- Equation 1

V3(sin A) = -(V4)(sin 25)  ---- Equation 2

V3^2 + V4^2 = 36   ---- Equation 3

with three unknowns (V3, V4 and angle A). I trust that you know your basic Algebra to solve for three unknowns given three equations.

Good luck.