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

Answer 1

Answer: (D)

Here $x_1+x_2=s$

and $t_1+t_2=t$
We have from $A$ to $C$
$v^2=u^2+2f{x}_1$
$v^2=0+2f{x}_1$
$\Rightarrow x_1=\frac{v^2}{2f}$...(i)
and $v=u+ft$
$\Rightarrow v=0+f{t}_1$
$\Rightarrow t_1=\frac{v}{f}$...(ii)
From $C$ to $B,v^2=u^2+2fs$
$v^{\prime 2}=v^2-2r{x}_2$
$0=v^2-2r{x}_2$
$\Rightarrow x_2=\frac{v^2}{2r}$...(iii)
and $v^{\prime }=v-r{t}_2$
$\Rightarrow 0=v-r{t}_2$
$\Rightarrow t_2=\frac{v}{r}$...(iv)
On adding Eqs. (i) and (iii)
$x_1+x_2=\frac{v^2}{2}\left(\frac{1}{f}+\frac{1}{r}\right)$
$2s=v^2\left(\frac{1}{f}+\frac{1}{r}\right)$...(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)
$\Rightarrow \frac{t^2}{2s}=\frac{v^2{\left(\frac{1}{f}+\frac{1}{r}\right)}^2}{v^2\left(\frac{1}{f}+\frac{1}{r}\right)}$
$=\left(\frac{1}{f}+\frac{1}{r}\right)$
$\Rightarrow t=\sqrt{2s\left(\frac{1}{f}+\frac{1}{r}\right)}$