Question

Figure shows a quarter circle. A particle moves from $B$ to $O$ with $5m/s$, from $O$ to $A$ with $7m/s$. With what speed the particle should move from $A$ to $B$ so that average speed for total journey is $6m/s$?

• A. $\frac{210}{35-\frac{4}{\pi }}m/s$
• B. $\frac{210}{31-\frac{4}{\pi }}m/s$
• C. $\frac{210}{38-\frac{4}{\pi }}m/s$
• D. $\frac{205}{35-\frac{4}{\pi }}m/s$

Answer 1

The correct option is A $\frac{210}{35-\frac{4}{\pi }}m/s$

Let the distance $AB$ be traversed with a speed $v$
Average speed $=\frac{\text{Total distance}}{\text{Total time}}$
$6=\frac{BO+OA+AB}{\frac{BO}{5}+\frac{OA}{7}+\frac{AB}{v}}$
$6=\frac{R+R+\frac{\pi R}{2}}{\frac{R}{5}+\frac{R}{7}+\frac{\pi R}{2v}}$
$6=\frac{2+\frac{\pi }{2}}{\frac{12}{35}+\frac{\pi }{2v}}$
$\Rightarrow \frac{72}{35}+\frac{3\pi }{v}=2+\frac{\pi }{2}$

On solving,
$v=\frac{210}{35-\frac{4}{\pi }}$