Question

An object's velocity is given by v(t)=0.12t^2 + 8t m/sec. At t=10, the object's position is x=500m. What is the equation for x(t)?

Answer 1

`x(t)=0.04t^3+4t^2+60`

Explanation:

Recall that if you're given a velocity function, the integral of the velocity function gives you a class of position functions ("a class" because there will be the arbitrary constant of integration, meaning that the solution is general).

So,

`v=0.12t^2+8t`

Integrate:

`x(t)=int(0.12t^2+8t)dt`

`x(t)=0.04t^3+4t^2+C`

Do not forget that constant of integration, it is central to this problem and we need to find it.

`x(10)=500,` so

`500=0.04(10^3)+4(10^2)+C`

`500=40+400+C`

`500=440+C`

`C=60`

The position is thus given by

`x(t)=0.04t^3+4t^2+60`