Answer to Question #170246 in Calculus for Jake

Question

At time t=0 a car is moving 6m/s then smoothly accelerate so that the acceleration after t seconds is a a(t)=3t m/s

Answer 1

$a(t)=\dfrac{dv}{dt}$

$dv=adt$

$\int dv=\int adt$ $v=\int adt$

Given $a=3t$

$v=\int(3t)dt$

$v(t)=\dfrac{3}{2}t^2+C_1$

If $v(0)=6$

$v(0)=\dfrac{3}{2}(0)^2+C_1=>C_1=6$

$v(t)=\dfrac{3}{2}t^2+6$

$v(t)=\dfrac{ds}{dt}$

$ds=vdt$

$\int ds=\int vdt$ $s=\int vdt$ $s=\int(\dfrac{3}{2}t^2+6)dt$

$s(t)=\dfrac{1}{2}t^3+6t+s(0)$

If $s(0)=0$

$s(t)=\dfrac{1}{2}t^3+6t$