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
$t_0$, $v_0$, $r_0$, $\Delta r$
Acceleration of the vehicle is described by:
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.