A billiard ball is dropped from a height of 64 feet. Use the position function s (t) = - 16? 2 + ?0? + ?0 to answer the following. a. Determine the position function s (t), the velocity function v (t), and the acceleration function a (t). b. What is the velocity of the ball at impact?

s (t) = - 16*t^2 + 64

v (t) = - 32*t

a (t) = - 32 ft/s^2

v (t) = 64 ft/s ... At impact

Explanation:

Given:-

- The height of the billiard ball t = 0, h = 64 ft.

- The position function of an object under gravity is given by:

s (t) = - 16*t^2 + v_o*t + s_o

Find:-

a. Determine the position function s (t),

b. the velocity function v (t),

c. the acceleration function a (t).

d. What is the velocity of the ball at impact?

Solution:-

- To determine the position function we must initialize our problem and use the given general equation.

- s (t) is the position of the billiard ball from the ground at time t. So when t = 0, then s (t) = h. Hence, we have:

s (t) = s_o = h = 64 ft

- Similarly we know that v_o is the initial velocity of the ball. Since, the ball was dropped we say that the initial velocity v_o = 0. Hence, the position of the ball from ground is given by following expression:

s (t) = - 16*t^2 + 64

- To find the velocity expression v (t) we will take the time derivative of the position expression s (t) as follows:

v (t) = d s (t) / dt

v (t) = - 16*2*t + 0

v (t) = - 32*t ft/s

- Similarly, the expression for acceleration a (t) is given by the time derivative of the velocity expression v (t) as follows:

a (t) = d v (t) / dt

a (t) = - 32*t

a (t) = - 32 ft/s^2

- The velocity of ball at impact can be determined by evaluating s (t) = 0 and find the value for time t. Then that time t can be substituted in the velocity expression v (t) for final velocity. Or we could use the following 3rd kinematic equation as follows:

v (t) ^2 - 0^2 = 2*a (t) * s_o

v (t) ^2 = 2 * (32) * (64)

v (t) = 64 ft/s

- The ball has a velocity of 64 ft/s at impact!