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*.

Practice the material and concepts discussed in Chapter 2.