Chapter 4.1, Problem 64E

Escape Velocity The minimum velocity required for an object to escape Earth’s gravitational pull is obtained from the solution of the equation

${\displaystyle \int vdv=-GM{\displaystyle \int \frac{1}{y^2}dy}}$ where v is the velocity of the object projected from Earth, y is the distance from the center of Earth, G is the gravitational constant, and M is the mass of Earth. Show that v and v are related by the equation

$v^2=v_0^2+2GM\left(\frac{1}{y}-\frac{1}{R}\right)$ where v₀ is the initial velocity of the object and K is the radius of Earth.