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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.

This set covers the material discussed in Chapter 8.