The first example follows the trajectory of a projectile launched from the north pole at an angle of elevation of 45 degrees. The program finds the latitude at which it lands, its impact angle, and its impact velocity.

The second example plots orbital motion under the influence of central
forces that vary as the *n*th power of the distance from the force
center. One can choose from six powers: *n* = 1 (spring),
*n* = 0 (constant), *n* = -1, *n* = -2 (gravity),
*n* = -3, and *n* = -5.
The orbiting particle starts off moving tangentially with a velocity of
0.488 in units where a velocity of 1.0 results in a circular orbit with
period 2p. For a force-law with *n* > -3
the orbit is drawn for a time of 47.3 in the units used, and the final position
is shown in magenta. The program determines Rmin, the distance of closest
approach to the force-centre, and the apsidal angle, the angle between Rmax
and Rmin. The spring and gravity produce orbits that close, and the initial
velocity was chosen to give the *n* = -1 orbit 16 lobes after 11
revolutions (to make a pretty picture).

Values of *n* more negative than -2 produce circular orbits for an
initial tangential velocity of 1.0, but quite different orbits for other
initial velocities. For *n* = -3 the orbit is an inward spiral about the
force-centre with an initial velocity of 0.999 (it would be an outward spiral
with an initial velocity of 1.001). For *n* = -5 the orbit is a circle
that passes through the force centre (the lower part of
the circle is drawn by reversing the direction of the initial velocity). The
analytic problem is usually worded: *If a particle moves in a circular
orbit passing through the centre of the force causing the motion, show that
the attraction varies as the inverse 5th power of the distance from the
force centre*. The initial velocity used in the applet, 1/sqrt(2), comes
from the analytic solution to this problem.

Although Newton's Law generates the central-force orbit directly, the nature
of the orbit is more clearly understood by making use of the fact that the
angular momentum is a constant of the motion. That is what is done in the
section on *Energy Conservation and Integration* under the heading
Conservation of Angular Momentum: Central-Force Motion.
When written in terms of the angular momentum, the tangential kinetic energy
varies inversely as *r* squared and can be treated as a centrifugal potential.
For forces with *n* = -2 or greater, the *r* motion is an
oscillation in a potential well, but for *n* = -5 the centrifugal
potential is overwhelmed, and the orbit reaches the force centre. For a
central force with *n* = -3, the potential has the same
*r*-dependence as the centrifugal potential, so any perturbation of a
circular orbit will cause either an inward or an outward spiral.

In the third example, an attractive force that varies inversely as the square of distance (gravity) is exerted between two or three bodies of comparable mass. The Feynman algorithm must be extended to deal with the motion of more than one body, but the steps in the calculation remain the same.

For two bodies there is an analytic solution that introduces center-of-mass coordinates, then treats the motion of a reduced-mass particle attracted to a fixed force center. These steps are not needed in the numerical solution, but it is instructive to view the orbits in a frame moving along with the centre of mass as well as in the lab frame. The program allows this option. The particles have a mass ratio of 3 to 2, the lighter particle starts out to the left, and vectors indicate the initial velocities. The motion obviously appears much simpler when viewed in the centre-of-mass frame.

For three bodies there is no analytic solution, but the numeric solution is no more difficult than it is for two bodies. There is, however, a new problem: two of the bodies are very likely to collide. As they approach each other, the forces become extremely large, and a very short time interval is needed to give accurate results. The program sets a minimum separation beyond which a collision is deemed to have taken place. It then uses a time interval small enough that no significant change occurs when it is cut in half. The third body has a mass equal to one-third of that of the most massive body, and it is initially at rest in the lab frame. (This ensures that the centre of mass moves in the same direction as it does for two bodies). For three bodies the motion is no simpler when viewed in the centre-of-mass frame. If one of the bodies is small enough that it does not affect the motion of the two larger bodies (the restricted three-body problem) the motion does have some interesting features. We study these in the section on The Trojan Asteroids.

The fourth example deals with Rutherford scattering. Scattering is the
term used to describe unbound orbits in central force fields (attractive or
repulsive). In Rutherford scattering, the force is the
repulsive Coulomb force felt by an a-particle as
it approaches the nucleus of a heavy atom. If the atom is heavy enough, the
nucleus can be treated as a fixed force center, and the Feynman algorithm
from the benchmark program in the second example can be used directly. The
incident trajectory is specified by two parameters: *a*, the distance of
closest approach in a head-on collision, and *b*, the impact parameter
(the distance of closest approach along the incident trajectory if there is
no interaction). In the example, the incident beam has a circular
cross-section of radius *b* = 2.5*a*. The starting point for the
numerical calculation is 40*a* from the scattering center (twice the
distance shown in the plot). The program offers three choices: (a) trajectories
for impact parameters stepped uniformly across the beam, (b) trajectories for
the same number of incident particles distributed randomly across the area of
the beam, and (c) a histogram versus scattering angle (in ten-degree
increments) for 500 randomly
distributed incident particles. In (c) the angular distribution
calculated by analytic methods is shown (in blue) for two seconds before
the particle beam is turned on. The analytic treatment evaluates the
integral in the *Apsidal Angles* section and finds that the impact
parameter and scattering angle are related by *tan*(q/2) = *a*/2*b*. (We will use *z* for this expression -
an expression that predicts, for example, that no particles in our beam
will be scattered through less than 22.6 degrees.) The angular distribution
calculated using this expression, the Rutherford scattering formula, is
usually written in terms of the inverse fourth power of a sine function, but
when all the particles are being counted it is more convenient to write the
angular distribution as proportional to
(1+*z*^{2})/*z*^{3}. This distribution function
predicts that 60 of the 500 incident particles will be
scattered through more than 60 degrees.