Motion of a body given acceleration as a function of velocity I'm having trouble figuring this out. I'm trying to determine the final time and final velocity for the motion of a vehicle over a given distance along a straight line.
$t$ is time
$v$ is velocity
$a$ is acceleration
$r$ is displacement
$m$ and $c$ are constant
Known:
$t_0$, $v_0$, $r_0$, $\Delta r$
Acceleration of the vehicle is described by:
$a=\frac{f(v)-c}{m}$
I want to derive expressions for:
$\Delta t$, and $v$
$f(v)$ is a function which returns the force which the engine can apply at a given velocity. This is essentially a lookup against a table of values from the vehicle manufacturer. I can perform an accumulation over this if required.
I'm currently estimating a result to this problem by iteratively calculating using small intervals of time.
Any help would be greatly appreciated.

As user7530 says, you could integrate numerically by using a first order  Euler method to integrate numerically
\begin{align*} r_{i+1} &= r_i + h v_i\\ v_{i+1} &= v_i + h a_{i+1} \\
a_{i+1} &  = \frac{f(v_i) -c}{m}\\ t_{i+1} &= t_i + h \end{align*}
until $r_i-r_0 > \Delta r$.
But you could use different methods to integrate numerically.  One way to test whether it's worth using a more complicated method would be to replace your looked up acceleration function with something that has a nice formmula for f(v), and then compare the numerical result with the exact result.
