Aims:

To discuss the motion of bodies under the action of forces.

To introduce the idea of a conservative force and the associated potential.

To discuss the particular example of inverse square law forces and the nature of orbits.

Objectives:

By the end of this course you should be able to:

• Connect change of momentum with impulse and solve for the motion of bodies acted on by an impulse.

• Know what is meant by the terms time-dependent force, position-dependent force and velocity-dependent force.

• Solve equations of motion for simple time-dependent forces.

• Solve equations of motion for simple velocity-dependent forces.

• Describe the difference between conservative and non-conservative forces.

• Give examples of conservative forces in Nature.

• Derive the potential energy in a uniform gravitational field, a stretched spring and in an inverse square law field.

• Define total energy and know under what conditions it is constant.

• Use potential and energy methods to solve for the motion of bodies under conservative forces.

• Derive a conservative force from its potential energy.

• Calculate the grad of a function in 3-D cartesian coordinates and 2-D polar coordinates.

• Derive the radial and tangential equations of motion in plane polar coordinates.

• Discuss the motion in a circular orbit under an inverse square law force including the use of reduced mass.

• Analyse non-circular orbits under an inverse square law of force using conservation laws.

• Compare motion under gravitational and electrostatic forces.

• Derive and calculate the maximum and minimum distances of approach and the escape velocity.

• Describe the principles of transfer orbits and the slingshot effect.

• Write down Kepler's Laws of planetary motion.

• Prove Kepler's 2nd Law for any central force.

• Write down the cartesian and polar equations for an ellipse.

• Define focus, eccentricity, major and minor axes of an ellipse and show how the other conic sections are described by the same polar equation.

• Write down the polar equations of motion for a particle in a general orbit.

• Outline the steps required to show that the orbits under an inverse square law of force are conic section solutions.

• Describe the different possible orbits and their features.

• Know how to modify the expression for the total energy of a circular orbit to give the total energy of an elliptical orbit.

• Know how to modify the expression relating the period and the radius of a circular orbit so that it is applicable to an elliptical orbit i.e. the general expression of Kepler's 3rd Law.

• Solve problems similar to those done in lectures and given on the examples sheet.