A 7.00-kg bowling ball moving at 10.0 m/s collides with a 1.60-kg bowling pin, s

Question

A 7.00-kg bowling ball moving at 10.0 m/s collides with a 1.60-kg bowling pin, scattering it with a speed of 8.00 m/s and at an angle of 33.5° with respect to the initial direction of the bowling ball.

(a) Calculate the final velocity (magnitude and direction) of the bowling ball.

Magnitude= 8.48 m/s

Direction= ?

(b) Ignoring rotation, what was the original kinetic energy of the bowling ball before the collision?
J

(c) Ignoring rotation, what is the final kinetic energy of the system of the bowling ball and pin after the collision?

J

I was able to find the magnitude, however I am struggling to find the direction as well as part b and c. It would also really help me to see your work. Thank you!!

Explanation / Answer

Let’s assume that the direction of the ball’s initial velocity is the positive x direction. Let’s determine its initial momentum.

M = 7 * 10 = 70
For the pin, final x momentum = 1.6 * 8 * cos 33.5 = 12.8 * cos 33.5
For the pin, final y momentum = 1.6 * 8 * cos 33.5 = 12.8 * sin 33.5
To determine the x component of the ball’s final momentum, subtract 12.8 * cos 33.5 from 70.

x = 70 – 12.8 * cos 33.5 = 76.2852
vx = 76.2852 ÷ 7 = 10.898 m/s

Since the ball was initially moving in the positive x direction, it had no y momentum.
For the sum of the y momentums to be 0, its y momentum must be -12.8 * sin 33.5.

vy = -(12.8/7) * sin 33.5
This is approximately negative -1.5929 m/s
To determine the magnitude of its final velocity, use the following equation.

Speed = (x^2 + y^2)
x^2 = 10.898^2
y^2 = (-12.8/7.0 * sin 33.5)^2 = 11.013^2 + 10.898^2 = 121.303
Speed = 121.303
This is approximately 11.014 m/s.

Tan = y ÷ x = -11.013 ÷ 10.898
The angle is approximately negative -45.3°. This is -45.3° clockwise from the direction of the ball’s initial velocity.

For the ball, original KE = ½ * 7 * 10^2 = 350 J
For the ball, final KE = ½ * 7 * 121.303 = 424.5605 J
For pin, final KE = ½ * 1.6 * 8^2 = 51.2 J

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