Question

Two particles move in the same straight line starting at the same moment from the same point in the same direction. The first moves with constant velocity $u$ and the second starts from rest with constant acceleration $f$. Then

• A. They will be at the greatest distance at the end of time $\frac{u}{2f}$ from the start
• B. They will be at the greatest distance at the end of time $\frac{u}{f}$ from the start
• C. Thier greatest distance is $\frac{u^2}{2f}$
• D. Thier greatest distance is $\frac{u^2}{2f}$

Answer 1

Answer: (B), (C)

The correct options are
B They will be at the greatest distance at the end of time $\frac{u}{f}$ from the start
C Thier greatest distance is $\frac{u^2}{2f}$

Let $S$ is the distance between the particles at time $t$
Distance covered by particle $1$ in time $t$ is
$x_1=ut$
Distance covered by particle 2 in time 2 is
$x_2=\frac{1}{2}f{t}^2$
Now. $S=x_1-x_2=ut-\frac{1}{2}f{t}^2...\left(1\right)$
For maximum distance $\frac{ds}{dt}=0$
$\Rightarrow u-\frac{1}{2}\times 2ft=0$
$\Rightarrow u-ft=0$
$\Rightarrow t=\frac{u}{f}$
$\Rightarrow \text{They will be at the greatest distance at the end of time}\frac{u}{f}\text{from the start.}$
Put value of $t$ in equation (1)

$S=u.\frac{u}{f}-\frac{1}{2}.\frac{u^2}{f^2}.f$
$\Rightarrow S=\frac{u^2}{2f}$
Hence, The greatest distance is $\frac{u^2}{2f}$