Answer
$a)$ $a=-kv$
$b)$ $c$ is the initial velocity of the object
$c)$ $t=\frac{\ln 2}{k}$
Work Step by Step
$a)$
$$a=v'(t)=(ce^{-kt})'=-kce^{-kt}=-kv$$
The factor of proportionality is $-k$.
$b)$
$$v(0)=ce^{-k \cdot 0}=ce^{0}=c$$ so $c$ is the initial velocity of the object.
$c)$
$$v(t)=\frac{c}{2}$$
$$ce^{-kt}=\frac{c}{2}$$
$$e^{-kt}=\frac{1}{2}$$
$$\ln(e^{-kt})=\ln(\frac{1}{2})$$
$$-kt=-\ln 2$$
$$t=\frac{\ln 2}{k}$$
