A spacecraft of mass is first brought into an orbit around the earth. The earth (together with the spacecraft) orbits the sun in a near circular orbit with radius ( is the mean distance between the earth and the sun; it is about 150 million km).

(a) What is the speed (in m/s) of the earth in its orbit of radius around the sun with a mass
We want the spacecraft to fall into the sun. One way to do this is to fire the rocket in a direction opposite to the earth's orbital motion to reduce the spacecraft's speed to zero (relative to the sun).

(b) What is the total impulse that would have to be given by the rocket to the spacecraft to accomplish this? You may ignore the effect of the earth's gravitation as well as the orbital speed of the spacecraft around the earth as the latter is much smaller than the speed of the earth around the sun. Thus, you may assume that the spacecraft, before the rocket is fired, has the same speed in its orbit around the sun as the earth. Express your answer in terms of and (enter m for and v_0 for ).



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We will now show that there is a more economical way of doing this (i.e., a much smaller rocket can do the job). By means of a brief rocket burn the spacecraft is first put into an elliptical orbit around the sun; the boost is provided tangentially to the earth's circular orbit around the sun (see figure). The aphelion of the new orbit is at a distance from the sun. At aphelion the spacecraft is given a backward impulse to reduce its speed to zero (relative to the sun) so that it will subsequently fall into the sun.

(c) Calculate the impulse required at the first rocket burn (the boost). Express your answer in terms of , and (enter I_0 for , R for and r for ).



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(d) What is the speed of the spacecraft at aphelion? Express your answer in terms of , and (enter v_0 for , R for and r for ).



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(e) Calculate the impulse required at the second rocket burn (at aphelion). Express your answer in terms of , and (enter I_0 for , R for and r for ).



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(f) Compare the impulse under b) with the sum of the impulses under c) and e) (i.e find ), and convince yourself that the latter procedure is more economical. Express your answer in terms of , and (enter I_0 for , R for and r for ).



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