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A spaceship approaches the Moon (mass = ...

A spaceship approaches the Moon (mass `= M` and radius `= R` along a parabolic path which is almost tangential to its surface. At the moment of the maximum approach, the brake rocket is fired to convert the spaceship into a satellite of the Moon. Find the change in speed.

Text Solution

Verified by Experts

The correct Answer is:
`(sqrt(2)-1)`
If `v` the velocity at the vertex of the parabola, then `v` is also the escape velocity. It will follow the parabolic path ever to return to the Moon. Now `v_("escape")=sqrt((2GM)/R)`
`=/_\v=f_("final")=v_("initial")=v_("orbit")-v_("escape")`
`implies /_\v=sqrt((GM)/R)=-sqrt((2GM)/R)`
`=-sqrt((GM)/R)(sqrt(2)-1)`

The negaive sign means the speed has to be decreased.
Therefore, the required change in the speed is
`sqrt((GM)/R)(sqrt(2)-1)`
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