Q. Three particles start from the origin at the same time, one with a velocity $v_{1}$ along $x$-axis, the second along the $y$-axis with a velocity $v_{2}$ and the third along $x = y$ line. The velocity of the third so that the three may always lie on the same line is
A) $\frac{v_{1}v_{2}}{v_{1} + v_{2}}$
B) $\frac{\sqrt{2}v_{1}v_{2}}{v_{1} + v_{2}}$
C) $\frac{\sqrt{3}v_{1}v_{2}}{v_{1} + v_{2}}$
D) Zero
Explanation
(b) Let time interval be chosen as 1 s
$\frac{PA}{PB} = \frac{OA}{OB} = \frac{v_{x}}{v_{y}}$
So, $P(x,y)$divides $AB$ is the ratio $v_{x}:v_{y}$.
Using section formula,
$x = \frac{v_{x} \times 0 + v_{y} \times v_{x}}{v_{x} + v_{y}} = \frac{v_{x}v_{y}}{v_{x} + v_{y}}$
[image:3_image158.emf]
$y = \frac{v_{x}v_{y}{+ v}_{y} \times 0}{v_{x} + v_{y}} = \frac{v_{x}v_{y}}{v_{x} + v_{y}}$
$v = \sqrt{x^{2} + y^{2}} = \sqrt{2}\frac{v_{x}v_{y}}{v_{x} + v_{y}}$
Now, replace $v_{x}$ by $v_{1}$ and $v_{y}$ by $v_{2}$.
$v = \frac{\sqrt{2}v_{1}v_{2}}{v_{1} + v_{2}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.