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5 June, 12:29

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?

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  1. 5 June, 12:34
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    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!
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