Chapter 10. Escape speed (10-10)

Review

The initial speed \(v_i\) for which Equation 10-9 is true is called the escape speed, which we denote by the symbol \(v_{\mathrm{escape}}\) . This is the minimum speed at which an object must be launched from Earth’s surface to escape to infinity. To find its value, replace \(v_i\) in Equation 10-9 with \(v_{\mathrm{escape}}\), add \(Gm_{\mathrm{Earth}}m/R_{\mathrm{Earth}}\) to both sides and multiply both sides by \(2/m\).

\(\frac{1}{2}mv_{\mathrm{escape}}^2 = \frac{Gm_{\mathrm{Earth}}m}{R_{\mathrm{Earth}}}\)

\(v_{\mathrm{escape}}^2 = \frac{2Gm_{\mathrm{Earth}}}{R_{\mathrm{Earth}}}\)

Finally, take the square root of both sides: