Question

A small block of superdense material has mass $2\times 10^{24} \, kg$ . It is at a height $h < < R$ . It falls towards the earth. Find its speed when it is at a height $h/2$ .

• A. $\sqrt{\frac{2 g h}{3}}$
• B. $\sqrt{\frac{3 g h}{4}}$
• C. $\sqrt{\frac{3 g h}{5}}$
• D. $\sqrt{\frac{g h}{2}}$

Answer 1

Answer: (B)

Mass of earth is, $M=6\times 10^{24} \, kg$
Mass of superdense material $m=2\times 10^{24}kg=\frac{M}{3}$
From momentum conservation, $MV_{1}=\frac{M}{3}V_{2}$
From energy conservation $V_{1}=\frac{V_{2}}{3}$
$\frac{1}{2}MV_{1}^{2}+\frac{1}{2}\left(\frac{M}{3}\right)V_{2}^{2}=\frac{G M \left(\frac{M}{3}\right)}{\frac{h}{2} + R}-\frac{G M \left(\frac{M}{3}\right)}{h + R}$
$\therefore \, \, v_{2}=\sqrt{\frac{3 g h}{4}}$