Projectile Motion III: Central-Force Motion

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.

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The second example plots orbital motion under the influence of central forces that vary as the nth 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.

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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.

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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 = 2.5a. The starting point for the numerical calculation is 40a 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/2b. (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+z2)/z3. This distribution function predicts that 60 of the 500 incident particles will be scattered through more than 60 degrees.

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