Kinematics with Calculus

$v = \frac{ds}{dt}$, $a = \frac{dv}{dt} = \frac{d^2s}{dt^2}$

$s = \int v\,dt$, $v = \int a\,dt$

$v = 6t - t^2$. When at rest? $t(6-t) = 0$ → $t = 0$ or $t = 6$.

Max velocity: $a = 6 - 2t = 0$ → $t = 3$. $v_{max} = 9$.

Displacement from $t=0$ to $t=6$: $\int_0^6 (6t-t^2)\,dt = [3t^2 - \frac{t^3}{3}]_0^6 = 108 - 72 = 36$.