The work required to launch an object from the surface of Earth to outer space is given by where R = 6370 km is the approximate radius of Earth, F(x) = GMm/x² is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of Earth, m is the mass of the object, and GM = 4 × 10¹⁴ m³/s².

a. Find the work required to launch an object in terms of m.

b. What escape velocity v_e is required to give the object a kinetic energy 1/2 mv ² _e equal to W?

c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light, c = 300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R ≤ 2GM/c². For Earth to be a black hole, what would its radius need to be?

W = JR F(x) dx,