Gravity and maximum height question.

A ball is thrown straight upward and rises to a maximimum height of 16m above its launch point. At what height above its launch point has the speed of the ball decreased to 1/2 its initial value?

Explanation that a hole through the center of the earth executes simple harmonic motion.

The gravitational force on a body located a distance R from the center of a uniform spherical mass is due solely to the mass lying at distance rclick for more

Proving that the motion models the given equation and finding the constant values in a situation where a mass is sliding on a rotating rod.

A particle of mass m is free to slide on a thin rod. The rod rotates in a plane about one end at constant angular velocity w. Show that the motion is given by r=Ae^(-yt)+Be^(yt), where y is a constant which you must find and A and B are arbitrary constants. Neglect gravity. Show that for a particular choice of initial conditions, (r(t=0)) and (v(t=0)) it is possible to obtain a solution such that r decreases continually in time, but that for any other choice r will eventually increase. (Exclude cases where the bead hits the origin.)

Exploding Projectile

An instrument carrying projectile accidentally explodes at the top of its trajectory. The horizontal distance between the launch point and the point of explosion is L. The projectile breaks into pieces which fly apart horizonally. The larger piece has three times the mass of the smaller piece. To the surprise of the scientist in charge, the smaller piece returns to earth at the lauching station. How far away does the larger piece land? Neglect air resistance and effects due to the earth's curvature

Conservation of momentum in a situation with men jumping off a flatcar.

N men, each with mass, m, stand on a railway flatcar of mass, M. They jump off one end of the flatcar with velocity u relative to the car. The car rolls in the opposite direction without friction. a) What is the final velocity of the flatcar if all the men jump at the same time? b) What is the final velocity of the flatcar if all the men jump off one at a time? (Can be left in form of a sum of terms) Does case a or case b yield the largest final velocity of the flat car? Can you give a simple physical explanation for your answer?