Question

The velocity acquired by a body when it falls through a height $h$ is $v$. If it further falls through a very small height $x$, the increase in velocity is approximately

• A. $\frac{vx}{2h}$
• B. $\frac{2vx}{h}$
• C. $\frac{2v}{xh}$
• D. $0.01v$

Answer 1

The correct option is A $\frac{vx}{2h}$


Using Third law of motion,
$v=\sqrt{2gh}.......\left(1\right)$
$v^{\prime 2}=v^2+2gx$
$v^{\prime 2}=(\sqrt{2gh}{)}^2+2gx=2g(h+x)v^{\prime }=\sqrt{2g(h+x)}....(2)$
From $\left(1\right)$ and $\left(2\right)$
$\frac{v^{\prime }}{v}=\sqrt{\frac{2g(h+x)}{2gh}}=\sqrt{\frac{h+x}{h}}$
$\Rightarrow \frac{v^{\prime }}{v}={(1+\frac{x}{h})}^{\frac{1}{2}}\text{Using Binomial approximation,}\Rightarrow v^{\prime }=1+\frac{x}{2h}$
$\Rightarrow v^{\prime }=(1+\frac{x}{2h})v$
Change in velocity $=(v^{\prime }-v)=v+\frac{xv}{2h}-v=\frac{xv}{2h}$