Question

A company sends a satellite up to 4.0 Earth radii from the center of the Earth and then launches it at a speed of 7423 m/s. Their hope is that it will go into a circular orbit at that distance.

a) Can a satellite be in circular orbit at that distance and speed without using any kind of propulsion?

b) How fast should the satellite go in order to orbit at this distance? [...] they tell the satellite to fire a set of emergency rockets for 2.0 minutes, thereby correcting its speed.

c) What average angular acceleration do we see as the satellite changes its speed? (magnitude)

Answer 1

Let the mass of satellite be "m" and that of Earth be "M_e".

then

Radius of orbit will be R= 4.0(Radius of Earth)= 4R_e.

We know that value of Radius of Earth as

Now when the satellite moves around the Earth then the centripetal force acting on the satellite will be exactly balanced by the Gravitational attractive force between satellite and Earth.

    

where v= velocity of satellite

G= Gravitational Constant,

  

  

  

  

    

SInce the speed of satellite at the time of launch is much larger than the needed for it. Thus the satellite will not be in circular orbit of given radius with that much speed.

b). Required Linear velocity,

Then Required Angular velocity,

  

Initial linear velocity of satellite,

Then Initial Angular velocity,

  

Time taken,

Let angular acceleration be , then by using equations of motion for rotational motion,

  

  

  

(ANS) (negative sign shows that it will be decceleration)