Question

The mysterious visitor that appears in the enchanting story The Little Prince was said to come from a planet that "was scarcely any larger than a house!" Assume that the mass per unit volume of the planet is about that of Earth and that the planet does not appreciably spin. Take the diameter of the planet to be 13 m. Approximate
(a) the free-fall acceleration on the planet's surface
(b) the escape speed from the planet.

Answer 1

Answer:

Part a)

$g = 1 \times 10^{-5} m/s^2$

Part b)

$v = 0.011 m/s$

Explanation:

Part a)

As we know that the acceleration due to gravity is given as

$g = \frac{GM}{R^2}$

$g = \frac{G(\frac{4}{3}\pi R^3 \rho)}{R^2}$

$g = \frac{4}{3}\pi \rho G R$

now we know that

$\rho = \frac{M}{\frac{4}{3}\pi R^3}$

$\rho = \frac{5.98 \times 10^{24}}{\frac{4}{3}\pi (6.37 \times 10^6)^3}$

$\rho = 5523.2 kg/m^3$

now we have

$g = \frac{4}{3}\pi (5523.2) (6.67 \times 10^{-11})(6.5)$

$g = 1 \times 10^{-5} m/s^2$

Part b)

As we know that the escape speed is given as

$v = \sqrt{\frac{2GM}{R}}$

$v = \sqrt{2gR}$

$v = \sqrt{2(1\times 10^{-5})(6.5)}$

$v = 0.011 m/s$