As we all know it the Newton's second law of motion describes the relationship between force and acceleration of a body as diectly proportional. Now lets say a body in motion and is of mass of 9kg and its acceleration is $a($t)=-9$t$. we are to find its displacement $y$ as a funtion of time $t$.
I know from calculus that when you integrate acceleration you get velocity and when you integrate velocity you get displacement. i followed suit but i want to see if this a valid way of going by it or... perphaps its wrong
Let $y(t)$ be the displacement on time $ t$. Then
$$a=\frac{d^2y}{dt^2}=-9t$$ after a first integration, we find
$$\frac{dy}{dt}=\int_{t_0}^t(-9u)du$$
$$=\Bigl[-\frac{9u^2}{2}\Bigr]_{t_0}^t$$ $$=-\frac{9t^2}{2}+V_0$$
the second integration gives
$$y(t)=\int_{t_0}^t(-\frac 92u^2+V_0)du$$
$$=\Bigl[-\frac{3}{2}u^3+V_0u\Bigr]_{t_0}^t$$
$$=-\frac 32 t^3+V_0t+Y_0$$
$Y_0$ and $ V_0$ are respectively the position and the velocity at $t=0$.