The escape speed is defined to be the minimum speed with which an object of mass

Question

The escape speed is defined to be the minimum speed with which an object of mass m must move to escape from the gravitational attraction of a much larger object, such as a planet of total mass M. The escape speed is a function of the distance of the object from the center of the planet R, but unless otherwise specified this distance is taken to be the radius of the planet because it addresses the question "How fast does my rocket have to go to escape from the surface of the planet?"

Part A:

The key to making a concise mathematical definition of escape speed is to consider the energy. If an object is launched at its escape speed, what is the total mechanical energy Emech of the object at a very large (i.e., infinite) distance from the planet? Follow the usual convention and take the gravitational potential energy to be zero at very large distances.

Part B:

Find the escape speed vesc for an object of mass m that is initially at a distance R from the center of a planet of mass M. Assume that RRplanet, the radius of the planet, and ignore air resistance.

Express the escape speed in terms of R, M, m, and G, the universal gravitational constant.

Explanation / Answer

a)

Etotal = 0

b)

vesc = sqrt(2 * G * M /R)

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