Chapter 13, Problem 66AP

Find the speed needed to escape from the solar system starting from the surface of Earth. Assume there are no other bodies involved and do not account for the fact that Earth is moving in its orbit. [Hint: Equation 13.6 does not apply. Use Equation 13.5 and include the potential energy of both Earth and the Sun.

Substituting the values for Earth’s mass and radius directly into Equation 13.6, we obtain

   $v_{esc}=\sqrt{\frac{2GM}{R}}=\sqrt{\frac{2\left(6.67\times {10}^{-11}N\cdot m^2/k{g}^2\right)\left(5.96\times {10}^{24} kg\right)}{\left(6.37\times {10}^{6}m\right)}}=1.12\times {10}^{4}m/s$

That is about 11 km/s or 25,000 mph. To escape the Sun, starting from Earth’s orbit, we use $R=R_{ES}=1.50\times {10}^{11}m$ and $M_{Sum}=1.99\times {10}^{30}\text{kg}$ . The result is $v_{\text{esc}}\text{=}4.21\times {10}^4\text{m}/\text{s}$ or about 42 km/s.

We have $\frac{1}{2}m{v}_{{}^{esc}}^2-\frac{GMm}{R}=\frac{1}{2}m{0}^2-\frac{GMm}{\infty }=0$

Solving for the escape velocity,