1.
Show that the areal velocity is constant for a particle moving
under the influence of an attractive force, given by *F(r*) = -*kr*. Calculate the time average of the
kinetic and the potential energy.

2.
A communication satellite is in a circular orbit of radius *R* around the earth. Its velocity is *v*. Its engine accidentally fires, giving
the satellite an outward radial velocity *v* in addition to its original velocity.

a. Calculate the ratio of the new energy and angular momentum to the old.

b.
Describe the subsequent motion of the satellite and plot *T*(*r*), *U*(*r*),
and *E*(*r*) after the engine fires.

Note: for a
circular orbit, *T* = - *U*/2.

3.
A particle of unit mass moves from infinity along a straight
line that, if continued, would allow it to pass a distance *b*√2 from a point *P*. If the
particle is attracted toward *P* with a force varying as *k*/*r*^{5}, and if the angular momentum about *P* is (√*k*)/*b*, show that the trajectory is given by

4.
A particle moves in an elliptical orbit in an inverse-square-law
central-force field. If the ratio
of the maximum angular velocity to the minimum angular velocity of the particle
in its orbit is *n*, then show that the
eccentricity of the orbit

5. Consider the family of orbits in a central potential for which the total energy is a constant. Show that if a stable circular orbit exists, the angular momentum associated with this orbit is larger than that for any other orbit in the family.

Practice the material and concepts discussed in Chapter 8.