1. A satellite of mass m orbits a planet of mass M and radius R as shown below.
(The diagram is not to scale.)
1. A satellite of mass m orbits a planet of mass M and radius R as shown below.
(The diagram is not to scale.)
The radius of the circular orbit of the satellite is x. The planet may be assumed to behave as a point mass with its mass concentrated at its centre.
(a) Deduce that the linear speed v of the satellite in its orbit is given by the expression
where G is the gravitational field constant
(b)
(i) Derive expressions, in terms of m, G, M and x, for the kinetic energy of the satellite
and for the gravitational potential energy of the satellite.
- Kinetic energy:
(ii) Deduce an expression for the total energy of the satellite. [2]
The satellite is moved into an orbit closer to the planet where there is friction with the planet's atmosphere.
(c)
(i) State the effect of these frictional forces on the total energy of the satellite.
(ii) Apply your equation in (b)(ii) to deduce that, as a result of this friction, the radius of the orbit will change continuously.
(iii) Describe the effect of this change in orbital radius on the speed of the satellite.
(iv) The frictional forces will change as the orbit of the satellite changes. Suggest and explain the effect on the motion of the satellite of these changing frictional forces.
2.
(a) Determine the ammount of energy that is required by a 40.0 kg satellite to launch it from the earths surface into a geostationary orbit. (Ignore any effects of the rotation of the earth.)
(b) Determine the amount of energy needed as in (a) above if the effect of the earths rotation was considered and the satellite was launched from the equator.
(c) How much chemical energy would be needed for the energy required to get the satellite in orbit (in part b)if the rocket had an average energy efficiency of 40%?
(d) Is this the total amount of energy needed to launch? Discuss.