At $t = 0$, a particle starts at rest and moves along a line in such a way that, at time $t$, its acceleration is $24\,t^2 \,\, ft/s^2$. Through how many feet does the particle move during the first $2$ seconds?
I was given this problem and I don't know how to solve it. I know it has to do with derivatives etc, I just don't know how to do these questions. I am not asking for the solution - just what are these kinds of problems called and where can I find more practice?
Note that the acceleration is time-varying. The speed at $t$ is obtained via the integral,
$$v(t) = \int_0^t a(s)ds= \int _0^t 24s^2ds=8t^3$$
Then, the displacement at $t=2$ is
$$\int_0^2 v(t)dt = \int_0^2 8t^3dt $$