A cue ball traveling at 0.70 m/s hits a stationary 8-ball, which moves off with a speed of 0.23 m/s at an angle of 39° relative to the cue ball's initial direction. Assuming that the balls have equal masses and the collision is inelastic, at what angle will the cue ball be deflected
To determine the angle at which the cue ball will be deflected, we need to analyze the conservation of momentum and the conservation of kinetic energy in the collision.
1. Conservation of Momentum:
In an inelastic collision, the total momentum before the collision is equal to the total momentum after the collision. Since the masses of both balls are equal, we can write the equation as:
(mass of cue ball × velocity of cue ball) = (mass of 8-ball × velocity of 8-ball)
Let's assign some variables:
- Mass of the cue ball = m
- Mass of the 8-ball = m
- Initial velocity of the cue ball = v
- Final velocity of the 8-ball = v8
- Angle of 8-ball relative to cue ball's initial direction = θ8
The momentum equation becomes:
m * v = m * v8 * cos(θ8) -- equation (1)
2. Conservation of Kinetic Energy:
In an inelastic collision, the total kinetic energy before the collision is not conserved. However, we can use the concept of kinetic energy loss to analyze the angle of deflection of the cue ball.
The kinetic energy loss is equal to the initial kinetic energy minus the final kinetic energy. Since both balls have equal masses, we can write it as:
0.5 * m * v^2 - 0.5 * m * v8^2 = Kinetic Energy Loss -- equation (2)
Now, let's solve both equations simultaneously to find the angle of deflection (θdef) of the cue ball:
Step 1: Use equation (1) to express v8 in terms of v and θ8:
v8 = v * cos(θ8) -- equation (3)
Step 2: Substitute equation (3) into equation (2):
0.5 * m * v^2 - 0.5 * m * (v * cos(θ8))^2 = Kinetic Energy Loss
Step 3: Rearrange and simplify the equation:
0.5 * m * v^2 - 0.5 * m * v^2 * cos^2(θ8) = Kinetic Energy Loss
0.5 * m * v^2 * (1 - cos^2(θ8)) = Kinetic Energy Loss
0.5 * m * v^2 * sin^2(θ8) = Kinetic Energy Loss
Step 4: Solve for sin^2(θ8):
sin^2(θ8) = (2 * Kinetic Energy Loss) / (m * v^2) -- equation (4)
Step 5: Take the square root of both sides to find sin(θ8):
sin(θ8) = sqrt((2 * Kinetic Energy Loss) / (m * v^2))
Step 6: Calculate θ8 by taking the inverse sine:
θ8 = arcsin(sqrt((2 * Kinetic Energy Loss) / (m * v^2)))
Finally, you can substitute the given values of Kinetic Energy Loss, m, and v to find the angle of deflection (θdef) of the cue ball.