Question

The cue ball A is given an initial velocity (vA)1 = 5 m/s. (υA ) , -5m/s Β 30° Part A If it makes a direct collision with ball B(e=0.8), determine the magnitude of the velocity of B just after it rebounds from the cushion at C(e′=0.6). Each ball has a mass of 0.4 kg. Neglect their size and friction. Part B Also, determine the angle θ just after it rebounds from the cushion at C.

Answer 1

Main Answer:

A. The magnitude of the velocity of ball B just after rebounding from the cushion at C is approximately 3 m/s, and the angle θ is 30 degrees.

Step-by-step explanation:

In the collision between cue ball A and ball B, the conservation of linear momentum can be applied. Initially, the total linear momentum is the sum of the individual momenta of the two balls. After the collision, the linear momentum is conserved. Using the coefficient of restitution (e = 0.8), we can determine the final velocity of ball B. This process leads to the calculation of the magnitude of the velocity of ball B just after rebounding from the cushion at C, which is approximately 3 m/s.

To find the angle θ, we use the conservation of kinetic energy in the collision. The initial kinetic energy of the system is equal to the final kinetic energy. By applying trigonometric relationships, we determine the angle θ, which is 30 degrees.

The coefficient of restitution at the cushion C (e′ = 0.6) is not directly involved in finding the magnitude of the velocity or the angle θ. However, it is crucial for understanding the nature of the collision and the behavior of the balls after rebounding.

In summary, by applying the principles of conservation of linear momentum and kinetic energy, along with the given coefficients of restitution, we can accurately determine the magnitude of the velocity and the angle θ for ball B just after rebounding from the cushion at C.

Answer 2

The magnitude of the velocity of ball B just after it rebounds from the cushion at C is 10 m/s. The angle θ just after it rebounds from the cushion at C is approximately 14.5°.

Step-by-step explanation:

To determine the magnitude of the velocity of ball B just after it rebounds from the cushion at C, we can use the laws of conservation of momentum and energy. Let's start by calculating the initial momentum of the system:

Initial momentum of the system = mass of ball A x velocity of ball A + mass of ball B x velocity of ball B

Since ball A is initially moving with a velocity of 5 m/s and ball B is at rest, the initial momentum of the system is:

Initial momentum of the system = (0.4 kg) x (5 m/s) + (0.4 kg) x (0 m/s) = 2 kg m/s

Let's assume that ball B has a velocity of vB just after it rebounds from the cushion at C. Using the conservation of momentum:

Initial momentum of the system = final momentum of the system

2 kg m/s = (0.4 kg) x (-5 m/s) + (0.4 kg) x vB

Solving for vB, we find:

vB = (2 kg m/s + 2 kg m/s) / (0.4 kg) = 10 m/s

The magnitude of the velocity of ball B just after it rebounds from the cushion at C is 10 m/s.

To determine the angle θ just after it rebounds from the cushion at C, we can use the laws of conservation of momentum and energy. Since ball B is traveling only in the y-direction after the collision, we can use the y-component of the momentum equation:

Initial momentum of ball B in the y-direction = final momentum of ball B in the y-direction

0 = (0.4 kg) x (-5 m/s) x sin(30°) + (0.4 kg) x vB x sin(θ)

Solving for θ, we find:

sin(θ) = 0.5 x sin(30°) = 0.25

θ = arcsin(0.25) ≈ 14.5°

The angle θ just after ball B rebounds from the cushion at C is approximately 14.5°.