Question

Use integration to find the position function for the given velocity function and initial condition. (Rubric 10 marks) \[ v(t)=3 t^{3}+30 t^{2}+5 ; s(0)=3 \]

Answer 1

`\displaystyle s(t)=(3)/(4)t^3+10t^3+5t+3`

Explanation:

Integrate v(t) with respect to time

`\displaystyle \int(3t^3+30t^2+5)\,dt\\\\=(3)/(4)t^4+10t^3+5t+C`

Plug-in initial condition to get C

`\displaystyle s(0)=(3)/(4)(0)^3+10(0)^3+5(0)+C\\\\3=C`

Thus, the position function is
`\displaystyle s(t)=(3)/(4)t^3+10t^3+5t+3` given the velocity function and initial condition.