In a game of pool, the 0.165 kg cue ball is hit towards the 0.155kg eight ball, which is stationary. The cue ball travels 6.2 m/s forward and after...
impact, rolls away at angle of 40.0 degrees counterclockwise from its initial direction, with a velocity of 3.7 m/s. What are the velocity and direction of the eight ball?
The speed and direction of the ball that was initially stationary can be determined using the law of conservation of momentum. According to the law, the total momentum of system of bodies undergoing collision initially is equal to the total momentum of the bodies after collision.
Here we have a cue ball that collides with an eight ball. The mass of the cue ball is 0.165 kg and it is initially traveling at 6.2 m/s. The mass of the eight ball is 0.155 kg and it is initially stationary.
The total momentum of the system is 0.165*6.2 = 1.023 kg*m/s in the forward direction.
After collision the cue ball rolls away at a velocity of 3.7 m/s in a direction 40 degrees counter-clockwise from the initial direction.
Let the velocity of the eight ball after collision be v in a direction forming an angle x with the original direction of the cue ball in the clockwise direction.
We can divide the momentum of the cue ball after impact into its components, one in the original direction of motion and one in the direction forming an angle of 90 degrees in the counter-clockwise direction. The magnitude of component in the original direction of motion is 3.7*0.165*cos 40 and the component in the direction forming an angle of 90 degrees counter-clockwise to the original direction is 3.7*0.165*sin 40.
By the law of conservation of momentum the velocity of the eight ball should have a component in the direction forming an angle of 90 degree clockwise to the original direction of motion equal to 3.7*0.165*sin 40. The component in the direction of the original motion should be 1.023 - 3.7*0.165*cos 40.
This gives the two components respectively as 0.3924 and 0.5553 kg*m/s. The components of the velocity are momentum divided by mass or 2.5317 and 3.5827 m/s respectively.
The two can be used to derive the final velocity of the ball and its direction. The final velocity is the square root of the squares of the two components or 4.387 and the angle formed by it in the clockwise direction is the arc tan of the value obtained when the normal component is divided by component in the original direction. The angle is 33.527 degrees.
The velocity of the eight ball is 4.387 m/s in a direction forming an angle of 33.527 degrees clockwise with the original direction of motion of the cue ball.