Answer:
s(t) = -16*t^2 + 64
v(t) = -32*t
a(t) = -32 ft/s^2
v(t) = 64 ft/s ... At impact
Step-by-step 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!