# How to determine the acceleration and position as a function of time

An object is losing mass at a rate k in kg/s. The object is acted on by a force F. Determine the acceleration and position as a function of time.

I know the answer for the position function is

$$x(t) =\left (\dfrac{Fm_0}{k^2}\right)\left[\left(\dfrac{-k}{m_0}\right)t - ln\left(1-\left(\dfrac{k}{m_0}\right)t\right)\right]$$

but I'm not entirely sure how to get there. If anyone could just help me out by listing a few equations that would relate the rate of loss of mass to force I would much appreciate it.

Recall Newton's 2nd law in the form $F=\frac{dp}{dt}$, where $p$ is momentum, namely $p(t)=m(t) v(t)$. Reformulating the equation gives you $F=\frac{dm}{dt}v+\frac{dv}{dt}m$. Using $m(t)={m}_{0}-kt$ yields the differential equation $$F=-kv+\stackrel{.}{v}\left({m}_{0}-kt\right)$$ Solve the equation to get $v(t)$, integrate to get $x(t)$ and differentiate to get $a(t)$.