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. their greatest distance is $\frac{u^{2}}{2f}$
• D. their greatest distance is $\frac{u^{2}}{f}$

Answer 1

Answer: (B), (C)

We know, $S=u t+\frac{1}{2} a t^{2}$…(i)
Condition for first,
As, velocity is constant $u=u, v=u$ (constant)
$\therefore a=0, S=u t \ldots$ (ii) [from Eq. (i)]
Condition for second $u=0, a=f$ (constant)
$\therefore S =\frac{1}{2} a t^{2} $
$ S=\frac{1}{2} f t^{2} \ldots $ (iii) [from Eq. (i)]
On differentiating Eqs. (ii) and (iii), we get
$\frac{d S}{d t}=u\,\,\,\,...$(iv)
and $\frac{d S}{d t}=\frac{1}{2} f(2 t)$
$\frac{d S}{d t}=f t$
$u=f t \,\,\,\,$ [from Eq. (iv)]
$t=\frac{u}{f}\,\,\,\,...$(v)
Thus, they will be at the greatest distance at the end of the $\frac{u}{f}$ from the start.
For greatest distance
$S=\frac{1}{2} f t^{2}$
Put $t=\frac{u}{f}$
$S=\frac{1}{2} f\left(\frac{u^{2}}{f^{2}}\right)$
$S=\frac{u^{2}}{2 f}$