Two skaters $\mathrm{P}$ and $\mathrm{Q}$ are skating towards each other. Skater $\mathrm{P}$ throws a ball towards $\mathrm{W}$ every $5 \mathrm{~s}$ such that it always leaves her hand with speed $2 \mathrm{~ms}^{-1}$ with respect to the ground. Consider two cases: $\mathrm{(I)}$ $\mathrm{P}$ runs with speed $1 \mathrm{~ms}^{-1}$ towards $\mathrm{Q}$ while $\mathrm{Q}$ remains stationary $\mathrm{(II)}$ $\mathrm{Q}$ runs with speed $1 \mathrm{~ms}^{-1}$ towards $\mathrm{P}$ while $\mathrm{P}$ remains stationary. Note that irrespective of speed of $\mathrm{P}$, ball always leaves $\mathrm{P's}$ hand with speed $2 \mathrm{~ms}^{-1}$ with respect to the ground. Ignore gravity. Balls will be received by $\mathrm{Q}$

• Correct answer
  A

  one every $2.5 \mathrm{~s}$ in case $\mathrm{(I)}$ and one every $3.3 \mathrm{~s}$ in case $\mathrm{(II)}$
• B

  one every $2 \mathrm{~s}$ in case $\mathrm{(I)}$ and one every $4 \mathrm{~s}$ in case $\mathrm{(II)}$
• C

  one every $3.3 \mathrm{~s}$ in case $\mathrm{(I)}$ and one every $2.5 \mathrm{~s}$ in case $\mathrm{(II)}$
• D

  one every $2.5 \mathrm{~s}$ in case $\mathrm{(I)}$ and one every $2.5 \mathrm{~s}$ in case $\mathrm{(II)}$

Solution

(I) Let initial distance between $\mathrm{P} \& \mathrm{Q}$ is $\mathrm{x}$
at $\mathrm{t}_{1}=\frac{\mathrm{x}}{2}$ a receive the ball. Next ball. $\begin{array}{l} t_{2}=\frac{x-5}{2}+5 \\ \Delta t=\frac{5}{2} \end{array}$ (II) in second case at $t=0$ $\mathrm{P}$ throws the ball.
$t_{1}=\frac{x}{3}$. Next ball $\begin{array}{l} t_{2}=\frac{x-5}{3}+5 \\ \Delta t=\frac{10}{3} \end{array}$

Hence, the correct answer is option: 'A'.