Question

A car is moving along a straight line, having initial speed of v0 (v sub 0) at t = 0. The car experiences a deceleration of a = -k/v where k
is a constant and v is the speed of the car at an instant. I need to find the speed of the car as a function of time.

Answer 1

We know that $a=\frac{dv}{dt}$ , so we have a differential equation to solve :

$\frac{dv}{dt} = -\frac{k}{v}$

$vdv = -kdt$

We integrate both sides from $t=0$ to $t=t$ :

$\int_{t=0}^{t=t} vdv = -\int_0^t kdt$

$(\frac{1}{2}v^2)|^{t=t}_{t=0} = -(kt)|^{t}_0$

$\frac{1}{2}(v(t)^2-v_0^2)=-kt$

$v(t)^2=v_0^2-2kt$

So we find our final solution :

$v(t)=\pm \sqrt{v_0^2-2kt}, 0\leq t\leq \frac{v_0^2}{2k}$

We take $+$ or $-$ depending on the sign of $v_0$ (we take the same sign that the sign of $v_0$ ) , as $v(t)$ must be continuous. We can also note that this solution exists only for some values of t, as for $\tau=\frac{v_0^2}{2k}, v(\tau)=0$ and thus the deceleration is not well defined at $t=\tau$ .