Description
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 F = −
αm
r
2
. If its orbit is an
ellipse with the semi-major axis a, derive the following relation between its velocity and r, a
v
2 = α
(
2
r
−
1
a
)
. (1)
What if the orbit is a hypberbola or parabola?
3. (35 points) Consider the scattering produced by a repulsive force F =
k
r
3
, show that the
cross section takes the form
σ(θ) = kπ2
2E
π − θ
θ
2
(2π − θ)
2 sin θ
. (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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