Question

If velocity of a particle moving along a straight line changes sinusoidally with time as shown in given graph. The average speed over time interval $t=0$ to $t=2(2n-1)$ second, n being any positive interget, is $\frac{x}{\pi }m/s$. Value of $x$ is (Answer upto 2 decimal digits)

Answer 1

Average speed = $\frac{\text{Total distance travelled}}{\text{Total time taken}}$
= $\frac{S}{2(2n-1)}$
Here, $\Delta t=2(2n-1)=4n-2=4(n-1)+2$

From the graph it is clear that time period $T$ is 4 $s$.
$\therefore \Delta t=(n-1)T+\frac{T}{2}$

Total distance travelled in one time period = $4A$, where $A$ is amplitude.

Therefore, total distance travelled in $\Delta t$ is
$S=(n-1)4A+2A=(2n-1)2A$

$\therefore <v>=\frac{(2n-1)2A}{2(2n-1)}=A$
and $\omega A=v_{max}=4$
$\Rightarrow \frac{2\pi }{4}A=4$
$\Rightarrow A=\frac{8}{\pi }$

$\therefore <v>=\frac{8}{\pi }m/s$