Question

A body travels a distance $s$ in $t$ seconds. It starts from rest and ends at rest. In the first part of the journey, it moves with constant acceleration $f$ and in the second part with constant retardation $r .$ The value of $t$ is given by :

• A. $2 s\left(\frac{1}{f}+\frac{1}{r}\right)$
• B. $\frac{2 s}{\frac{1}{f}+\frac{1}{r}}$
• C. $\sqrt{2 s(f+r)}$
• D. $\sqrt{2 s\left(\frac{1}{f}+\frac{1}{r}\right)}$

Answer 1

Answer: (D)

Here $x_1 + x_2 = s$

and $ t_{1}+t_{2}=t$
We have from $A$ to $C$
$v^{2} =u^{2}+2 f x_{1} $
$ v^{2} =0+2 f x_{1} $
$ \Rightarrow x_{1} =\frac{v^{2}}{2 f} $...(i)
and $ v=u+f t$
$ \Rightarrow v =0+f t_{1} $
$\Rightarrow t_{1} =\frac{v}{f} $...(ii)
From $C $ to $ B, v^{2} =u^{2}+2 f s$
$ v'^{ 2} =v^{2}-2 r x_{2} $
$ 0 =v^{2}-2 r x_{2} $
$ \Rightarrow x_{2} =\frac{v^{2}}{2 r} $...(iii)
and $v'=v-r t_{2}$
$\Rightarrow 0=v-r t_{2}$
$\Rightarrow t_{2}=\frac{v}{r}$...(iv)
On adding Eqs. (i) and (iii)
$x_{1}+x_{2} =\frac{v^{2}}{2}\left(\frac{1}{f}+\frac{1}{r}\right) $
$2 s =v^{2}\left(\frac{1}{f}+\frac{1}{r}\right)$...(v)
On adding Eqs. (ii) and (iv)
$\Rightarrow t_{1}+t_{2}=v\left(\frac{1}{f}+\frac{1}{r}\right)$
$\Rightarrow t =v\left(\frac{1}{f}+\frac{1}{r}\right) $
$\Rightarrow t^{2} =v^{2}\left(\frac{1}{f}+\frac{1}{r}\right)^{2}$...(iv)
On dividing Eqs. (vi) by (v)
$\Rightarrow \frac{t^{2}}{2 s} =\frac{v^{2}\left(\frac{1}{f}+\frac{1}{r}\right)^{2}}{v^{2}\left(\frac{1}{f}+\frac{1}{r}\right)} $
$=\left(\frac{1}{f}+\frac{1}{r}\right) $
$ \Rightarrow t =\sqrt{2 s\left(\frac{1}{f}+\frac{1}{r}\right)}$