Launching conditions of a satellite
Take 15 minutes to prepare this exercise.
Then, if you lack ideas to begin, look at the given clue and start searching for the solution.
A detailed solution is then proposed to you.
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A satellite is injected in orbit on M0. The distance between M0 and the center O of the Earth is .
Its initial velocity is . it is orthogonal to .
Express this velocity with , (mass of the Earth) and if the trajectory is a circle of radius .
In the following questions, we note this particular velocity .
If is the radius of the Earth, we note :
Show that the satellite will not escape the gravitational pull of the Earth and will not meet the Earth if :
What is the mechanical energy on a parabola ? On an ellipse ?
The satellite escapes the gravitational pull of the Earth if its mechanical energy cancels out (its motion is a parabola).
The velocity is such as :
If the satellite touches the Earth, its trajectory is an ellipse with a major axis of length :
The energy for an elliptical motion is :
This energy is equal to the one it had when it was launched :
We can deduce the corresponding value of :
Finally, the satellite does not espace the Earth's gravitational pull and does not meet the Earth if :