Problem 1:

Assume for this problem the Earth is a sphere of radius R and mass M.  An object of mass m enters Earth’s atmosphere at distance R’ > R from Earth’s center with speed v at angle a from the radial direction.  Ignoring any friction or air resistance, what angle b (from the radial direction) will it hit Earth’s surface at?

Problem 2:

A particle of mass m moves in a circular orbit of radius r  = a under the influence of the central attractive force
f(r) = -g exp(-br)/r²,
where g and b are positive constants.
(a)  What is the effective potential energy, U_eff(M,r), for radial motion in terms of r and the angular momentum M?  (Your answer may contain an integral.)
(b)  For what values of b will this orbit be stable?
(c)  What is the frequency of small radial oscillation about these stable circular orbits?

Problem 3:

This question is about an elliptical “transfer orbit” B from an inner circular orbit A to an outer circular orbit B.  The transfer starts at point P and is completed at point Q.  The transfer orbit is an ellipse which is tangent to A at point P and tangent to C at point Q.
(a)  Derive a formula for the relationship between v and r for circular orbits.  Is the speed in orbit C greater or less than the speed in orbit A?
(b)  For the transfer, should the satellite speed up or slow down at point P?
(c)  For the transfer, should the satellite speed up or slow down at point Q?

Problem 4:

A comet of mass m approaches the solar system with a velocity v₀, and if it had not been attracted towards the sun, it would have missed the sun by a distance d.  Calculate its minimum distance z from the sun as it passes through the solar system.  Make and state any reasonable simplifying assumptions.

Problem 5:

(a)  Derive the relationship between the impact parameter b and the scattering angle q 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 6:
A particle of mass m moves in a central force field such that its potential energy is given by V = krⁿ, where r is the distance from the center of force and k and n are constants.
(a)      Write down the Lagrangian for this system and determine the equations of motion in polar coordinates. 
(b)      Show that angular momentum is conserved for the system.
(c)      Find an expression for the total energy of the system that depends only on the radial variable.
(d)      Find the conditions (sign and magnitude of n and k) for a stable circular orbit by investigating the particle at stable equilibrium.

Problem 7:

Find the maximum time a comet (C) of mass m following a parabolic trajectory around the Sun (S) can spend within the orbit of the Earth (E).  Assume that the Earth’s orbit is circular and in the same plane as that of the comet.