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(R2 + 2GMR/v2).
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?
Consider a particle of mass m moving in the xy plane. The potential energy
U(x,y) = (k/2)/(x2 + y2), 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?
(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.
Three stars with masses m1, m2, and m3 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.