an overhead view of the path taken by a 0.162 kg cue ball as it bounces from a rail of a pool table. The ball's initial speed is 1.11 m/s, and the angle è1 is 64.7°. The bounce reverses the y component of the ball's velocity but does not alter the x component. What are (a) angle è2 and (b) the magnitude of the change in the ball's linear momentum? (The fact that the ball rolls is irrelevant to the problem.)
Since one component of velocity stays the same and the other only changes sign, the kinetic energy (KE) does not change. That means the magnitude of the linear momentum, which is sqrt(2*KE*M), doesn't change either. The angle of reflection will equal the angle of incidence, but will be the supplement of 64.7 degrees (115.3 degrees) when measured from a common set of axes.
To find the angle (è2) and the magnitude of the change in the ball's linear momentum, we can break down the problem into smaller steps. Let's go through the process step by step.
Step 1: Determine the initial velocity components
First, let's find the initial velocity components of the cue ball. We are given the initial speed (1.11 m/s) and the angle (è1 = 64.7°). To find the x and y components of the velocity, use the following equations:
Vx = V * cos(è1)
Vy = V * sin(è1)
Vx represents the velocity component in the x-direction, and Vy represents the velocity component in the y-direction.
Step 2: Determine the final velocity components
Since the ball bounces from the rail, the y-component of the velocity will reverse, while the x-component remains the same. This means that the final velocity components will be:
V'x = Vx
V'y = -Vy
Step 3: Find the angle (è2) of the final velocity
Now that we have the final velocity components, we can find the angle (è2) using the following equation:
tan(è2) = (V'y) / (V'x)
Step 4: Calculate the magnitude of the change in linear momentum
To calculate the magnitude of the change in linear momentum, we need to subtract the initial linear momentum from the final linear momentum. The linear momentum is given by:
p = m * v
where p is the linear momentum, m is the mass of the cue ball (0.162 kg), and v is the velocity.
The change in linear momentum (δp) is then:
δp = m * v' - m * v
where v' represents the final velocity and v represents the initial velocity. Take the absolute value of δp to find the magnitude.
By following these steps, you should be able to find the values for (a) angle è2 and (b) the magnitude of the change in the ball's linear momentum.