Problem

A rocket with mass M = 12.0 metric.tons is moving around the Moon in a circular orbit at t...

A rocket with mass M = 12.0 metric.tons is moving around the Moon in a circular orbit at the height of h = 100. km. The braking engine is activated for a short time to lower the orbital height so that the rocket can make a lunar landing. The velocity of the ejected gases is u =1.00 ⋅ 104 m/s. The Moon’s radius is RM = 1.74⋅103 km; the acceleration of gravity near the Moon’s surface is gM = 1.62 m/s2.

a) What amount of fuel will be used by the braking engine if it is activated at point A of the orbit and the rocket lands on the Moon at point B (see the left part of the figure)?

b) Suppose that, at point A, the rocket is given an impulse directed toward the center of the Moon, to put it on a trajectory that meets the Moon’s surface at point C (see the right part of the figure). What amount of fuel is needed in this case?

Step-by-Step Solution

Solution 1

Step #1 of 10

THINK:

The equation for the speed of the circular orbit is found by setting the force of gravity equal to the centripetal force. The conservation of momentum can be used to find the amount of fuel needed in each case.

Step #2 of 10

RESEARCH:

Speed of the rocket in orbit is