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

Question

The escape velocity 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 body, such as a planet of total mass M. The escape velocity is a function of the distance of the object from the center of the planetR, 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?" What is the total mechanical energy Etotal of the object at a very large (i.e., infinite) distance from the planet? Find the escape velocity ve 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 velocity in terms of R, M, m, andG, the universal gravitational constant.

Explanation / Answer

let R = distance of object from the planet's center
M = planet mass
m = object mass
G = universal gravitational constant

for an object to escape, total energy = 0 ... the object will have just enough kinetic energy to take it an infinite distance from the planet

U + K = 0

-GMm/R + (1/2)mv^2 = 0, where v = escape velocity

(1/2)mv^2 = GMm/R
v = sqrt(2GM/R)

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