Arocket lifts off the pad at cape canaveral. according to newton's law of gravitation, the force of gravity on the rocket is given by f(x) = − gmm x2 where m is the mass of the earth, m is the mass of the rocket, g is a universal constant, and x is the distance (in miles) between the rocket and the center of the earth. take the radius of the earth to be 4000 miles, so that x > 4000 miles. find the work, w1, done against gravity when the rocket rises 2000 miles.

The equation is the Law of Universal Gravitation. The gravitational constant G is equal to 6.67×10⁻¹¹ Nm²/kg². The mass of the Earth is 5.972 ×10²⁴ kg. Compared to the mass of the Earth, the mass of the rocket is negligible. So, we don't need to know the mass of the rocket. Substituting the values:

F = (6.67×10⁻¹¹ Nm²/kg²)(5.972 ×10²⁴ kg)/(4000 miles*(1.609 km/1mile))²
F = 9616423.08 N

The work is equal to
W = Fd
W = (9616423.08 N)(2000 miles*1.609 km/mile)
W = 9.095×10¹⁰ Joules