Newton's second law with variable mass Let's say there's a body that is being affected only by a constant force $F$, but the body is vaporizing at the speed $β$, which is equal to the loss of some mass every second. The initial speed of the body is 0. What would be a function for acceleration versus time and for speed versus time? The initial mass is $m_0$.
 A: \begin{align}
F&=\frac{dp}{dt}\\
&=\frac{d}{dt}(mv)\\
&=\dot mv+m\dot v\\
&=-\beta v+ma
\end{align}
Here I defined $\beta\equiv -\dot m$ as the mass evaporation rate to make $\beta$ positive. I assume $\beta$ is constant which gives $m(t)=m_0-\beta t$. This results in the following differential equation
$$(m_0-\beta t)\ddot x(t)-\beta\dot x(t)-F=0$$
Plugging that into Mathematica gives after some rearranging
$$x(t)=\frac F\beta \left(\frac\beta{m_0}\right)^{-1}\left[-t\frac\beta{m_0}-\log\left(1-t\frac\beta{m_0}\right)\right]$$
Here $F/\beta$ and $\beta/m_0$ are quantities that naturally popped up during the rearranging. They have units of $LT^{-1}$ and $T^{-1}$ respectively. Compare this solution to the case where $\beta=0$
$$x(t)=\frac{F}{2m_0}t^2$$
I plotted these solutions for a couple different values of $\beta/m_0$ where I kept $\frac F\beta \left(\frac\beta{m_0}\right)^{-1} =1$ and I included $\beta=0$ as a separate case because it has a different solution. This looks as follows

Note that the solutions stop existing if $t\geq(\beta/m_0)^{-1}$ which corresponds to the mass becoming less than zero, i.e. when the particle stops existing.

Mathematica code:
(* Solving the differential equations *)
DSolve[{(m0 - \[Beta] t) x''[t] - \[Beta] x'[t] - F == 0, x[0] == 0, 
  x'[0] == 0}, x, t]
DSolve[{(m0) x''[t] - F == 0, x[0] == 0, x'[0] == 0}, x, t]

(* input the simplified function with a==F/beta, b==beta/m0 *)
sol1[t_, a_, b_] := a (b)^-1 (-t b - Log[1 - t b])
sol2[t_, a_, b_] := (a b)/2 t^2

(* enforce a*b==1 *)
sol3[t_,x_]:=sol1[t,1/x,x]
sol4[t_,x_]:=sol2[t,1/x,x]

Plot[Evaluate[{ sol3[t, .5], sol3[t, 1], sol3[t, 4], 
   sol4[t, 1/1, 1]}], {t, 0, 1}, 
   FrameLabel -> {"t", "x"}, 
   Frame -> True, 
   PlotLegends -> {"\[Beta]/\!\(\*SubscriptBox[\(m\), \(0\)]\) = 0.5", 
   "\[Beta]/\!\(\*SubscriptBox[\(m\), \(0\)]\) = 1", 
   "\[Beta]/\!\(\*SubscriptBox[\(m\), \(0\)]\) = 4", 
   "\[Beta] = 0 (Constant mass)"}]

