Question

The escape velocity from the earth is $11.2km/s$. The escape velocity from a planet having twice the radius and the same mean density as the earth, is:

• A. $11.2km/s$
• B. $22.4km/s$
• C. $15.00km/s$
• D. $5.8km/s$

Answer 1

Answer: (B)

At a certain velocity of projection, the body will go out of the gravitational field to the earth and will never return, this initial velocity is called escape velocity
$v_e=\sqrt{\frac{2G{M}_e}{R_e}}$
where $G$ is gravitational constant, $M_e$ is mass of earth and $R_e$ is radius.
For planet $R_p=2R_e,d_p=d_e$
Also since earth is assumed spherical in shape its mass is given by
$M=\frac{4}{3}\pi R^3\times d$
$\therefore v_e=\sqrt{\frac{2G}{R_e}\times \frac{4}{3}\pi R^{3}d}$ (1)
$v_p=\sqrt{\frac{2G}{2R_e}\times \frac{4}{3}\pi {\left(2R_e\right)}^{3}d}$(2)
Dividing Eq. (1) by Eq. (2), we get
$\frac{v_e}{v_p}=\frac{1}{2}$
$\Rightarrow v_p=2v_e=2\times 11.2=22.4km/s$