A free-falling object is described by $h = 35 t + \frac { 1 } { 2 } a t ^ { 2 }$ , where h is the height above the ground, t is the amount of time the object is falling, and a is the acceleration. Solve the equation for a .
A) $a = \frac { 2 ( h - 35 t ) } { t }$
B) $a = \frac { 2 ( h + 35 t ) } { t ^ { 2 } }$
C) $a = \frac { 2 ( h - 35 t ) } { t ^ { 2 } }$
D) $a = \frac { h - 35 t } { t ^ { 2 } }$
E) $a = \frac { h - 35 t } { 2 t ^ { 2 } }$

Question

Quizplus · Accepted Answer

Firstly, taking the derivative of h with respect to t gives the velocity, v:

$$v = \frac{dh}{dt} = 35+at$$

Then, taking the derivative of v with respect to t gives the acceleration, a:

$$a = \frac{dv}{dt} = \frac{d}{dt}(35+at) = \frac{d}{dt}(35) + \frac{d}{dt}(at) = 0 + a = a$$

Therefore, the acceleration is just the coefficient of t in the equation for h:

$$a = \frac{2(h-35t)}{t^2}$$

which is choice C.

A Free-Falling Object Is Described By h=35t+12at2h = 35 t + \frac { 1 } { 2 } a t ^ { 2 }h=35t+21​at2

A Free-Falling Object Is Described By $h = 35 t + \frac { 1 } { 2 } a t ^ { 2 }$