Question

At t = 0, a particle starts at rest and moves along a line in such a way that, at time t, its acceleration is 24t 2 ft / s2. through how many feet does the particle move during the first 2 seconds?

Answer 1

The acceleration of the particle at time t is:
$a(t)=24t^2 ft/s^2$
The velocity of the particle at time t is given by the integral of the acceleration a(t):
$v(t)= \int a(t) \, dt = \int (24 t^2) dt=24 \frac{t^3}{3}=8t^3 ft/s$
and the position of the particle at time t is given by the integral of the velocity v(t):
$x(t)=\int v(t) = \int (8t^3)=8 \frac{t^4}{4}=2t^4 ft$

Assuming the particle starts from position x(0)=0 at t=0, the distance the particle covers in the first t=2 seconds can be found by substituting t=2 s in the equation of x(t):
$x(2 s)=2 t^4 = 2 (2s)^4=32 ft$