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uniform gravitational field and collision

1) Find the curl, delta x F, for the following forces:
(a) F = kr;
(b) F = (Ax, By2, Cz3);
(c) F = (Ay2, Bx, Cz),
where A, B, C and k are constants.

2) Verify that the gravitational force - GMmr/r2 on a point mass m at r, due to a fixed point mass M at the origin, is conservative and calculate the corresponding potential energy.

3) A mass m is in uniform gravitational field, which exerts the usual force F = mg vertically down, but with g varying with time, g = g(t). Choosing axes with y measured vertically up and defining U = mgy as usual, show that F = - delta U as usual, but, by differentiating E = (1/2)mv2 + U with respect to t, show that E is not conserved.

4) Verify the three equations:
that give x, y, z in terms of the spherical polar coordinates r, theta, phi.
(b) Find expressions r, theta, phi in terms of x, y, z.

5) Consider a head-on elastic collision between two particles. Prove that the relative velocity after the collision is equal and opposite to that before. That is, v1 - v2 = - (v'1 - v'2), where v1 and v2 are the initial velocities and v'1 and v'2 the corresponding final velocities.

6) A particle of mass m1 and speed v1 collides with a second particle of mass m2 at rest. If the collision is perfectly inelastic, what fraction of the kinetic energy is lost in the collision? Comment on your answer for the cases that m1 << m2 and that m2 << m1.

See attached file for full problem description.


Solution Summary

It finds the potential energy of the uniform gravitational field. It also finds the velocity and kientic energy after the collision. The solution is detailed and well presented.