Question

Solve the given question

If the acceleration varies with time as a equal to t cube find the average acceleration and average velocity for the time interval 2 sec to 5 sec ?

Answer 1

Assuming that the body starts from rest,
$a(t)=t^3\\ \Rightarrow \dfrac{dv}{dt}=t^3\\ \Rightarrow \int_{0}^{v}dv=\int_{0}^{t}t^3dt\\ \Rightarrow v(t)=\dfrac{t^4}{4}\\ \Rightarrow \dfrac{dx}{dt}=\dfrac{t^4}{4}\\ \Rightarrow \int_{0}^{s}dx=\int_{0}^{t}\dfrac{t^4}{4}\\ \Rightarrow s(t)=\dfrac{t^5}{20}\\ \langle a\rangle =\dfrac{\Delta v}{\Delta t}\\ =\dfrac{v(5)-v(2)}{5-2}\\ \langle v\rangle =\dfrac{\Delta v}{\Delta t}\\ =\dfrac{s(5)-s(2)}{5-2}$
I hope you can solve it from here.