Answer
Proof given below.
Work Step by Step
Define "up" as the positive direction of motion. Acceleration is the derivative of velocity
$a(t)=v'(t)=-32\ \ \mathrm{f}\mathrm{t}/\mathrm{s}^{2}$
$v(t)=\displaystyle \int(-32)dt=-32t+C\ \ \mathrm{f}\mathrm{t}/\mathrm{s}$
Given that $v(0)=v_{0}$
$v_{0}= -32(0) +C$
$C=v_{0}$
Thus,
$v(t)=-32t+v_{0}.$
At the highest point, the projectile momentarily has zero velocity.
This happens when
$v(t)=0$
$0=-32t+v_{0}$
$t=v_{0}/32$ seconds
