The work required to launch a spacecraft.

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Chapter 13, Problem 13.62P

(a)

To determine

The work required to launch a spacecraft.

(b)

To determine

The additional work required to move the spacecraft from low earth orbit to a very great distance from the earth.

(c)

To determine

The justification of the statement “In terms of energy, low earth orbit is halfway to the edge of the universe”.

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(a) Calculate how much work is required to launch a spacecraft of mass m from the surface of the earth (mass mE, radius RE) and place it in a circular low earth orbit—that is, an orbit whose altitude above the earth’s surface is much less than RE. (As an example, the International Space Station is in low earth orbit at an altitude of about 400 km, much less than RE = 6370 km.) Ignore the kinetic energy that the spacecraft has on the ground due to the earth’s rotation. (b) Calculate the minimum amount of additional work required to move the spacecraft from low earth orbit to a very great distance from the earth. Ignore the gravitational effects of the sun, the moon, and the other planets. (c) Justify the statement “In terms of energy, low earth orbit is halfway to the edge of the universe.”

The gravitational potential energy U(r) of a point mass m at a distance r from thecentre of a planet of mass M is given by U(r) = −GMm/r, where G is Newton’s Gravitational Constant. (a) What is the total energy of an asteroid of mass 3500 kg if it is moving directlytowards the centre of the Earth at a speed of 12 kms−1 at a distance of 10Earth radii from the centre of the Earth?(b) What is the kinetic energy of the asteroid when it hits the surface of the Earth?(c) What is the speed of the asteroid when it hits the surface of the Earth? Note: in parts (b) & (c) you may neglect any friction with Earth’s atmosphere; youmay take the mass of the Earth to be 5.97 × 1024 kg and the radius of the Earth tobe 6.37 × 106 m.

A satellite of mass m is in an elliptical orbit around the Earth, which has mass ME and radius RE. The orbital radius varies from the smallest value rA at point A to the largest value rB at point B. The satellite has a velocity vA at point A. Assume that the gravitational potential energy Ug = 0 when the satellite is at an infinite distance from the Earth. Present all the answers in terms of G, m, ME, RE, rA, rB, and vA. Derive an expression for the gravitational potential energy of the satellite as a function of distance r from the center of the Earth by using an appropriate definite integral. Determine the total mechanical energy of the satellite when it is at point A. Determine the angular momentum of the satellite with respect to the center of the Earth when it is at point A. Determine the velocity of the satellite when it is at point B.