Question

Solve using integration & explain briefly.
Example 0.17. The velocity of a particle is given by v = $v_0\sin \omega t$, where $v_0$ is constant and $\omega =2\pi /T$. Find the average velocity in time interval t = 0 to t = T/2.

Answer 1

Dear Student

$$\begin{gathered} v=v_o\sin \omega t \\ {v}_{avg}={\int }_0^{T/2}{v}_o\sin \omega tdt \\ {v}_{avg}=\frac{v_o}{\omega }{\left[-\cos \omega t\right]}_0^{T/2}{ \\ }_{avg}=\frac{v_o}{\omega }\left[-\cos \omega \frac{T}{2}+\cos 0\right]{ \\ }_{avg}=\frac{v_o}{\omega }\left[-\cos \left(\frac{2\mathrm{\pi }}{T}\right)\frac{T}{2}+1\right]{ \\ }_{avg}=\frac{v_o}{\omega }\left[-\mathrm{cos\pi }+1\right]{ \\ }_{avg}=\frac{v_o}{\omega }\left[-\left(-1\right)+1\right]=2\frac{v_o}{\omega } \end{gathered}$$
Regards