Question

If the radius of the earth is reduced by $1\%$ keeping the mass constant. The escape velocity will :

• A. increase by $0.5\%$
• B. decrease by $0.5\%$
• C. decrease by $11\%$
• D. remain same

Answer 1

Answer: (B)

decrease by $0.5\%$

Escape speed is $v_e=\sqrt{\frac{2GM}{R}}$
It is given that only radius is changing while mass is constant.
Differentiating $v_e$ with respect to $R$ gives
$\frac{\Delta v}{\Delta R}=\frac{-1}{2}\sqrt{2GM}\times R^{-3/2}=\frac{-1}{2}\sqrt{\frac{2GM}{R}}\times \frac{1}{R}=\frac{-1}{2}\frac{v}{R}$; where $\Delta$ indicates change.
$\therefore \frac{△v}{v}\times 100=\frac{-1}{2}\frac{\Delta R}{R}\times 100=-\frac{1}{2}\times (-1)=0.5$; $($ $-1$ because it reduced $)$