1. (20 points) Find the orbits of a point mass moving in a central force field F = −kr, where k is a positive constant. What if k is a negative constant?
2. (25 points) A point mass m moves in a central force field with. If its orbit is an ellipse with the semi-major axis a, derive the following relation between its velocity and r, a
. (1)
What if the orbit is a hypberbola or parabola?
3. (35 points) Consider the scattering produced by a repulsive force , show that the cross section takes the form
. (2)
4. (20 points) Show that for an antisymmetric 3 × 3 real matrix A, the matrix B = (1+A)(1−A)−1 is orthogonal, where 1 is the identity matrix.
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