Chapter 4, Problem 50Q

To determine

The escape speed for Earth and Mars.

The planet for which it would be easier to eject rocks.

Answer to Problem 50Q

The escape speed for Earth is 11.2 km/s and for Mars is 5 km/s

It is easy for rocks to escape from planet Mars.

Explanation of Solution

Given:

Gravitational constant, G= $6.67\times {10}^{-11}N{m}^2/k{g}^2$

Mass of Earth, $M=5.974\times {10}^{24}kg$

Radius of the Earth, $R=6.67\times {10}^{-11}N{m}^2/k{g}^2$

For Mars

Mass of Mars, $M=3.397\times {10}^{6}kg$

Radius of Mars, $R=6.418\times {10}^{23}m$

Formula Used:

An object needs particular speed to escape from the atmosphere which is called escape speed of the planet.

The escape speed is calculated as $V_{escape}=\sqrt{\frac{2GM}{R}}$

The escape speed is calculated as $V_{escape}=\sqrt{\frac{2GM}{R}}$

For Earth, substitute $6.67\times {10}^{-11}N{m}^2/k{g}^2$ for G (gravitation constant), $5.974\times {10}^{24}kg$ for M (mass) and $6.378\times {10}^{6}m$ for R (radius)

$\begin{array}{l}{V}_{escape}=\sqrt{\frac{2\left(6.67\times {10}^{-11}N{m}^2/k{g}^2\right)\left(5.974\times {10}^{24}kg\right)}{6.378\times {10}^{6}m}}\\ V_{escape}=\sqrt{\frac{2\left(6.67\times {10}^{-11}\left(\left(N\right)\left(\frac{kg.m/s^2}{N}\right)\right).m^2/k{g}^2\right)\left(5.974\times {10}^{24}kg\right)}{6.378\times {10}^{6}m}}\\ V_{escape}=\left(11,178m/s\right)\left(\frac{1km}{1000m}\right)\\ V_{escape}=11.178km/s\end{array}$

Round off the result to three significant figures the escape speed of the earth is 11.2 Km/s

Substitute $6.67\times {10}^{-11}N{m}^2/k{g}^2$ for G, $3.397\times {10}^{6}kg$ for M and $6.418\times {10}^{23}m$ for R

$\begin{array}{l}{V}_{escape}=\sqrt{\frac{2\left(6.67\times {10}^{-11}{m}^2/k{g}^2\right)\left(6.418\times {10}^{23}kg\right)}{3.397\times {10}^{6}m}}\\ V_{escape}=\sqrt{\frac{2\left(6.67\times {10}^{-11}\left(\left(N\right)\left(\frac{kg.m/s^2}{N}\right)\right).m^2/k{g}^2\right)\left(6.418\times {10}^{23}kg\right)}{3.397\times {10}^{6}m}}\\ V_{escape}=\left(5020m/s\right)\left(\frac{1km}{1000m}\right)\\ V_{escape}=5.02km/s\end{array}$

Round off the result to three significant figures the escape speed of mars is 5 km/s.

Conclusion:

From the above calculations, it is clear that it is easy for the rocks to escape from the planet Mars because it has a speed of 5 km/s only.