Suppose the position function for a free-falling object on a certain planet is given by $s ( t ) = - 13 t ^ { 3 } + v _ { 0 } t + s _ { 0 }$. A silver coin is dropped from the top of a building that is 1370 feet tall. Determine the velocity function for the coin.
A) $v ( t ) = - 13 t ^ { 3 } + 1370$
B) $v ( t ) = - 39 t ^ { 2 }$
C) $v ( t ) = - 39 t ^ { 3 } + 1370$
D) $v ( t ) = - 13 t ^ { 2 }$
E) $v ( t ) = - 3 t ^ { 4 }$

Question

Quizplus · Accepted Answer

The velocity function is the derivative of the position function with respect to time, $v(t) = s'(t)$. Differentiating $s(t) = -13t^3 + v_0t + s_0$ with respect to $t$ gives $v(t) = -39t^2 + v_0$. Since the coin is dropped, its initial velocity $v_0 = 0$, resulting in $v(t) = -39t^2$.

Suppose the Position Function for a Free-Falling Object on a Certain