Billiard Ball. We figured out in class that the place to hit a billiard ball so

Question

Billiard Ball. We figured out in class that the place to hit a billiard ball so that it immediately rolls without slipping is (2/5)R above the middle. If you instead hit it at (4/5)R above the middle, the ball will have some "forward English". If the ball initially leaves with translational velocity vo, find its eventual velocity when it starts rolling without slipping. Hint: This ball will start with non-zero translational velocity vo and non-zero rotational velocity wo. We don't know wo, but we can use the concept of impulse, Fdt = dp and Tdt dL, to determine the relationship between wo and vo since it is the same initial force F in contact for the same amount of time t that creates both. 3

Explanation / Answer

Fnet = - f = m a

a = - f / m

so v = v0 + a t = v0 - f t /m

and torque = I alpha

r f = (2 m r^2 / 5) alpha

alpha = 5 f / 2 m r

w = alpha t = 5 f t / 2 m r

it will start rolling without slipping,

v = w r

v0 - f t /m = r (5 f t / 2 m r )

v0 = ft / m + 5 ft / 2 m = 3.5 f t / m

t = m v0 / 3.5 f

v = v0 - f t /m = v0 - (f/m)( m v0 / 3.5 f )

v = v0 - v0/3.5

v = 5 v0 / 7 ......Ans

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