#### Problem 1:

A star of mass M and radius R is moving with
constant velocity **v** through a cloud of particles of density ρ.

If all the particles which collide with the star are trapped by it, show that the mass of
the star will increase at a rate

dM/dt = πρv(R^{2} + 2GMR/v^{2}).

#### Problem 2:

A satellite is in a circular orbit of radius r around an airless spherical planet of
radius R. An asteroid of equal mass falls radially towards the
planet, starting at zero velocity from a very large distance. The satellite and
the asteroid collide inelastically and stick together,
moving in a new orbit that just misses the planet's surface. What was the radius
r of the satellite's original circular orbit in terms of R?

#### Problem 3:

Consider a particle of mass m moving in the xy plane. The potential energy
function is

U(x,y) = (k/2)/(x^{2} + y^{2}), with k a positive constant.

(a) Find the equations of motion.

(b) Are there circular orbits? If yes, do they all have the same period?

(c) Is the total energy conserved?

#### Problem 4:

(a) Derive the relationship between the impact parameter b and the
scattering angle θ for Rutherford scattering of a projectile of mass m by a
fixed target particle of mass M.

(b) If 4 MeV alpha particles are incident
on a gold foil (Z = 79), calculate the impact parameter that would give a
deflection of 10 degrees.

(c) Explain what modifications of this calculation
must be made if the gold atom is allowed to recoil during the collision.

#### Problem 5:

Three stars with masses m_{1}, m_{2}, and m_{3} are
forming a peculiar triple-star system, where each of the stars is situated in
the corners of an equilateral triangle with a side length d. The stars are
attracting each other with gravitational forces. Determine the direction
and magnitude of the rotational velocity **ω** which will leave the relative
position of the three stars unchanged.