Chapter 11, Problem 26P

A space probe is fired as a projectile from the Earth’s surface with an initial speed of 2.00 × 10⁴ m/s. What will its speed be when it is very far from the Earth? Ignore atmospheric friction and the rotation of the Earth.

P11.26 $\begin{array}{l}{K}_i+U_i=K_f+U_f\\ \frac{1}{2}m{v}_i^2+G{M}_{E}m\left(\frac{1}{r_f}-\frac{1}{r_i}\right)=\frac{1}{2}m{v}_f^2\to \frac{1}{2}{v}_i^2+G{M}_E\left(0-\frac{1}{R_E}\right)=\frac{1}{2}{v}_f^2\\ \text{or}{v}_f^2=v_1^2-\frac{2G{M}_E}{R_E}\\ \text{and}{v}_f={\left(v_1^2-\frac{2G{M}_E}{R_E}\right)}^{1/2},\\ v_f={\left[{\left(2.00\times {10}^4\right)}^2-1.25\times {10}^8\right]}^{1/2}\text{m}/\text{s}=1.66\times {10}^4\text{m}/\text{s}\end{array}$