**Write the following text on the front cover of your homework assignment and
sign it. If the text is missing, 20 points will be subtracted from your
homework grade.**

1. A particle moves in a medium under the influence of a retarding force equal to

-*mk*(*v*^{3}*+a*^{2}*v*), where *k* and *a* are positive constants. Show that for any value of the initial
speed the particle will never move a distance greater than π/2*ka* and that the particle comes to rest
only when the time approaches infinity.

2. Show directly that the time rate of change of the angular momentum about the origin for a projectile fired from the origin is equal to the moment of the gravitational force (its torque) about the origin.

3.
A
particle of mass *m* = 1 kg is
subjected to a one-dimensional force

where *k* = 1 N/s and *a* = 0.5 s^{-1}. If the particle is initially at rest,
calculate and plot with the aid of a computer the position, speed, and acceleration
of the particle as a function of time.

4.
Consider
a particle moving in the region *x* > 0 under the influence of the potential

where *U*_{0} = 1 J and *a* = 2 m. Plot the potential, find the equilibrium
points, and determine whether they are maxima or minima.

5. Which of the following forces are conservative?

a.
*F _{x}* =

b.
*F _{x}* = -

c.

In these equations *a*, *b*,
and *c* are constants. For the conservative forces, find the
potential energy *U*.

This set covers the material discussed in Chapter 2.