Question

What must be the magnitude of the displacement of the automobile (from rest) after 4 seconds of travel if its acceleration as function time is a(t) = 2t m/s^2 along positive x-axis?

Answer 1

Let's assume, the car starts at $x_0=0$ with zero initial speed $v_0=0$. Then the velocity of the car is given as follows:

$$v(t) = v_0 + \int_0^ta(t)dt = \int_0^t2tdt = t^2\space m/s$$

The displacement is:

$$s(t)=x(t)-x_0 = x(t) = \int_0^tv(t)dt = \int_0^tt^2dt = \dfrac{t^3}{3}\space m$$

Substituting $t = 4s$, obtain:

$$s(4) = \dfrac{4^3}{3}= \dfrac{64}{3} m\approx 21.3m$$

Answer. Approximately 21.3 m.