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, $S=ut+\frac{1}{2}a{t}^2$…(i)
Condition for first,
As, velocity is constant $u=u,v=u$ (constant)
$\therefore a=0,S=ut\dots$ (ii) [from Eq. (i)]
Condition for second $u=0,a=f$ (constant)
$\therefore S=\frac{1}{2}a{t}^2$
$S=\frac{1}{2}f{t}^2\dots$ (iii) [from Eq. (i)]
On differentiating Eqs. (ii) and (iii), we get
$\frac{dS}{dt}=u...$(iv)
and $\frac{dS}{dt}=\frac{1}{2}f(2t)$
$\frac{dS}{dt}=ft$
$u=ft$ [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 $t=\frac{u}{f}$
$S=\frac{1}{2}f\left(\frac{u^2}{f^2}\right)$
$S=\frac{u^2}{2f}$