Question

The escape velocity on earth is 11.2 km/s. If the mass and radius of the Earth become half, then the new value of its orbital velocity is _________.

• A. $\frac{1.12}{\sqrt{2}}km/s$
• B. $\frac{5.6}{\sqrt{2}}km/s$
• C. $\frac{11.2}{\sqrt{2}}km/s$
• D. $11.2\left(\sqrt{2}\right)km/s$

Answer 1

The correct option is C

$\frac{11.2}{\sqrt{2}}km/s$

Step 1, Given data

Escape velocity on earth $V_e$=11.2 km/s

Mass and radius of the earth become half.

Step 2, Finding the new orbital velocity

If the escape velocity is 11.2 km/s the orbital velocity $V_0=\frac{V_e}{\sqrt{2}}=\frac{11.2}{\sqrt{2}}$km/s

Escape velocity is given as $V_e=\sqrt{\frac{2GM}{R}}$

Where, G = universal constant, M = Mass of the earth, R = Radius of the earth

Mass of the earth and radius become half

So, $V_e\text{'}=\sqrt{\frac{2G{\displaystyle \frac{M}{2}}}{{\displaystyle \frac{R}{2}}}}=\sqrt{\frac{2GM}{R}}=11.2km/s$

Now, New orbital velocity $V_o\text{'}=\frac{V_e\text{'}}{\sqrt{2}}=\frac{11.2}{\sqrt{2}}km/s$

Therefore the new orbital velocity is $\frac{11.2}{\sqrt{2}}km/s$

Hence the correct option is (C) $\frac{11.2}{\sqrt{2}}km/s$