Answer
a) $W$ = $\frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2$
b) $161.\bar{3}$ $ft-lb$
Work Step by Step
a)
$W$ = $\int_{x_1}^{x_2}f(x)dx$ = $\int_{t_1}^{t_2}f(s(t))v(t)dt$
$x$ = $s(t)$
$dx$ = $v(t)dt$
$W$ = $\int_{t_1}^{t_2}ma(t)v(t)dt$ = $\int_{v_1}^{v_2}mudu$
$u$ = $v(t)$
$du$ = $a(t)dt$
$W$ = $\left[\frac{1}{2}mu^2\right]_{v_1}^{v_2}$ = $\frac{1}{2}mv_2^2-\frac{1}{2}mv_1^2$
b)
mass = $\frac{12lb}{32ft/s^2}$ = $\frac{3}{8}$ slug
velocity = $\frac{20mi}{1h}\times\frac{5280ft}{1mi}\times\frac{1h}{3600s^2}$ = $\frac{88}{3}$ $ft/s^2$
so
$v_1$ = $0$
$v_2$ = $\frac{88}{3}$
$W$ = $\frac{1}{2}\times\frac{3}{8}\times(\frac{88}{3})^2-\frac{1}{2}\times\frac{3}{8}\times(0)^2$ = $\frac{484}{3}$ = $161.\bar{3}$ $ft-lb$
