Question

if a(t) = 6t - 12, v(0) = 3, and s(0) = 11, find s(1)

Answer 1

I assume a, v, and s denote acceleration, velocity, and position of some physical body, respectively.

Use the fundamental theorem of calculus.

The velocity function is

$v(t)=v(0)+\displaystyle\int_0^ta(u)\,\mathrm du$

$v(t)=3+\displaystyle\int_0^t(6u-12)\,\mathrm du$

$\implies v(t)=3t^2-12t+3$

and the position function is

$s(t)=s(0)+\displaystyle\int_0^tv(u)\,\mathrm du$

$s(t)=11+\displaystyle\int_0^t(3u^2-12u+3)\,\mathrm du$

$\implies s(1)=11+\displaystyle\int_0^1(3u^2-12u+3)\,\mathrm du$

$\implies s(1)=11+(u^3-6u^2+3u)\bigg|_0^1=\boxed{9}$