In the game of billiards, all the balls have approximately the same mass, about 0.20 kg. The cue ball moving west with a speed of 4.5m/s strikes another ball at rest. After the collision the cue ball moves at an angle of 55degree north of west and the other ball at an angle of 25 south of west. What is the speed of the cue ball and the other ball after the collision?
use momentum in x,ydirectrions, and conservation of energy.
To find the speed of the cue ball and the other ball after the collision, we can use the principles of conservation of momentum and conservation of kinetic energy.
1. Conservation of Momentum:
According to conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision.
Momentum = mass × velocity
Let's denote the final velocity of the cue ball as V1 and the final velocity of the other ball as V2. The initial velocity of the cue ball is 4.5 m/s moving west, and the initial velocity of the other ball is 0 m/s (at rest).
The momentum before the collision can be calculated as:
Initial Momentum = (mass of cue ball) × (initial velocity of cue ball) + (mass of other ball) × (initial velocity of other ball)
= (0.20 kg) × (-4.5 m/s) + (0.20 kg) × (0 m/s)
= -0.9 kg·m/s
The momentum after the collision can be calculated as:
Final Momentum = (mass of cue ball) × (final velocity of cue ball) + (mass of other ball) × (final velocity of other ball)
= (0.20 kg) × (V1) + (0.20 kg) × (V2)
Since momentum is conserved, we can equate the two expressions:
-0.9 kg·m/s = (0.20 kg) × (V1) + (0.20 kg) × (V2) ----(Equation 1)
2. Conservation of Kinetic Energy:
According to conservation of kinetic energy, the total kinetic energy before the collision should be equal to the total kinetic energy after the collision.
Kinetic Energy = (1/2) × mass × (velocity)^2
The initial kinetic energy of the cue ball can be calculated as:
Initial KE1 = (1/2) × (mass of cue ball) × (initial velocity of cue ball)^2
= (1/2) × (0.20 kg) × (4.5 m/s)^2
= 0.405 J
The initial kinetic energy of the other ball is zero since it's at rest.
The final kinetic energy of the cue ball can be calculated as:
Final KE1 = (1/2) × (mass of cue ball) × (final velocity of cue ball)^2
= (1/2) × (0.20 kg) × V1^2
The final kinetic energy of the other ball can be calculated as:
Final KE2 = (1/2) × (mass of other ball) × (final velocity of other ball)^2
= (1/2) × (0.20 kg) × V2^2
Since kinetic energy is conserved, we can equate the two expressions:
0.405 J = (1/2) × (0.20 kg) × V1^2 + (1/2) × (0.20 kg) × V2^2 ----(Equation 2)
Now, we have a system of two equations (Equation 1 and Equation 2) with two variables (V1 and V2). We can solve this system of equations to find the values of V1 and V2.
Solving Equation 1 and Equation 2 simultaneously will give us the final velocities of the cue ball and the other ball after the collision.