# Mechanical point

# 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.

If you have more questions, feel free to ask them on the forum.

A satellite is injected in orbit on M_{0}. The distance between M_{0} and the center O of the Earth is
.

Its initial velocity is . it is orthogonal to .

## Question

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 .

### Hint

### Solution

The second law of motion for a circular motion is :

## Question

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 :

### Hint

What is the mechanical energy on a parabola ? On an ellipse ?

### Solution

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 :