An unmanned spacecraft is in a circular orbit around a distant moon, observing t

Question

An unmanned spacecraft is in a circular orbit around a distant moon, observing the moon's surface from an altitude of 50.0  km . The mass of this particular moon is 7.150×1022  kg and its radius is 1.770×106 m ; the mass of the spacecraft is 921.0  kg . A group of scientists and engineers is controlling the spacecraft from a great distance; to their dismay, a programming error causes an on-board thruster to fire. The misprogrammed thruster fires in a direction opposite to the velocity of the spacecraft; the resulting thrust decreases the speed of the spacecraft by 28.032  m/s .

NOTE: This problem has a tolerance of only 0.5%, instead of the usual tolerance of 2%. Please enter your answers to this question to four significant figures rather than the usual three. Use 6.673×1011  Nm2/kg2 for the universal constant of gravitation in this problem.

Part A) What was the speed of the spacecraft before the unfortunate programming error? Also, what is the speed of the spacecraft immediately after the unfortunate programming error?

Part B)

Whereas the spacecraft had been in a circular orbit before the error occured, the controllers are able to determine that the spacecraft is now in an elliptical orbit. For this orbit, the maximum distance from the center of the moon is the orbital radius at which the programming error occurred, and the distance of closest approach (assuming all the mass of the moon were concentrated at a single point at the center of the moon) is alarmingly only only 1699.20  km , which is less than the radius of the moon!

Imagine that all the mass of the moon were concentrated at a single point at the center of the moon. If nothing is done to correct the orbit of the spacecraft, what would be the resulting speed of the spacecraft at its distance of closest approach to the imaginary concentrated mass? Your eventual task is to solve for this speed by two different methods.

What would be the kinetic energy of the spacecraft when at its distance of closest approach? Also, what would be the magnitude of the angular momentum of the spacecraft when at its distance of closest approach?

Give your answers as an ordered pair, with the kinetic energy first, followed by a comma, followed by the angular momentum.

Part C) What would be the resulting speed of the spacecraft at its distance of closest approach to the imaginary concentrated mass?

Part D) Of course, all the mass of the moon is not concentrated at its center. If nothing is done to correct the spacecraft's new orbit, it will crash into the surface of the moon. Your eventual task is to calculate the speed and direction with which it will crash. Begin by considering the possibility of using Conservation of Angular Momentum to find the speed.

The definition of angular momentum is L   r  × p , so that the magnitude of angular momentum is rpsin where is the angle between the position and momentum vectors. At the instant just before contact with the surface of the moon, what would be the angular momentum (magnitude only) of the spacecraft about the center of the moon? Also, what would be the magnitude of the position vector of the spacecraft, as measured from the center of the moon?

Give your answers as an ordered pair, with the magnitude of angular momentum first, followed by a comma, followed by the magnitude of the position vector. Give the magnitude of the position vector in kilometers.

Explanation / Answer

given, h = 50 km

r = 1732,000 m

m = 7.15*10^22 kg

Rm = 1.77*10^6 m

M = 921 kg

dv = 28.032 m/s

A) mv^2/r = GMm/r^2

mv^2 = GmM/r

v^2 = Gm/r

v = 2753493.07 m/s

b) rm = 1699200 m

from conservation of angular momentum

m*v*rm = m*v'*r'

r' = 1.77*10^6 m

v' = v - dv

v = 2.91*10^5 m/s

c) KE = 0.5M*v^2 = 39.223*10^12 J

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