Question

A stone is projected from the point on the ground in such a direction so as to hit a bird on the top of a telegraph post of height $h$ and then attain the maximum height $\frac{3h}{2}$ above the ground. If at the instant of projection, the bird were to fly away horizontally with uniform speed, find the ratio between horizontal velocities of the bird and stone if the stone still hits the bird while descending.

• A. $1:1$
• B. $1:2$
• C. $1:\sqrt{3}+1$
• D. $\sqrt{3}-1:1$

Answer 1

The correct option is D $\sqrt{3}-1:1$

Problem given in the question is illustrated with the help of diagram below:
Using the figure, we can say
$t_1+t_2=\frac{2u_y}{g}.......\left(i\right)$
For motion of the stone in vertical direction, we can say
$h=u_{y}t-\frac{1}{2}g{t}^2$
$\Rightarrow g{t}^2-2u_{y}t+2h=0.......\left(ii\right)$
$t_1\text{and}{t}_2$ are the roots of equation$\left(ii\right)$
$\Rightarrow t_1{t}_2=\frac{2h}{g}..........\left(iii\right)$
Also, $H=\frac{3h}{2}=\frac{u_y^2}{2g}\Rightarrow u_y=\sqrt{3gh}..........\left(iv\right)$
From $\left(i\right)$ and $\left(iv\right)$
$t_1+t_2=2\sqrt{\frac{3h}{g}}..........\left(v\right)$
and $t_1{t}_2=\frac{2h}{g}$
$\Rightarrow (t_2-t_1{)}^2=(t_1+t_2{)}^2-4t_1{t}_2$
$=\frac{12h}{g}-\frac{8h}{g}=\frac{4h}{g}$
$\Rightarrow t_2-t_1=2\sqrt{\frac{h}{g}}..........\left(vi\right)$
From $\left(v\right)$ and $\left(vi\right)$
$t_2=\sqrt{\frac{h}{g}}(\sqrt{3}+1)$
$AB=V_b{t}_2=u_x(t_2-t_1)$
$\therefore$ Ratios of horizontal velocities of bird and stone $=\frac{V_b}{u_x}=\frac{t_2-t_1}{t_2}=\frac{2}{\sqrt{3}+1}=\sqrt{3}-1$