The "galilean cannon" is a stack of elastic balls that are dropped from a height onto a flat surface. when they rebound, the smallest ball at the top can bounce to incredible heights. imagine a stack of n balls with masses m, , 2m, 3m, … etc. where > 1. the balls are dropped from a height ℎ.

a. assume all the collisions between balls are elastic. derive an expression for the height that the smallest ball bounces to, using m, ℎ, or some subset thereof

b. now assume that the ball-ball collisions are inelastic; when 2 balls collide, their total kinetic energy decreases by some factor i. e. = , with < 1. find such that the topmost ball rises back to the initial height ℎ. assume the bottom ball’s collision with the ground is perfectly elastic.

Some material consisting of a collection of microscopic objects is kept at a high temperature. a photon detector capable of detecting photon energies from infrared through ultraviolet observes photons emitted with energies of 0.3 ev, 0.5 ev, 0.8 ev, 2.0ev, 2.5ev, and 2.8ev. these are the only photon energies observed. (a) draw and label a possible energy-level diagram for one of the microscopic objects, which has four bound states. on the diagram, indicate the transitions corresponding to the emitted photons. explain briefly. (b) would a spring–mass model be a good model for these microscopic objects? why or why not? (c) the material is now cooled down to a very low temperature, and the photon detector stops detecting photon emissions. next, a beam of light with a continuous range of energies from infrared through ultraviolet shines on the material, and the photon detector observes the beam of light after it passes through the material. what photon energies in this beam of light are observed to be significantly reduced in intensity (“dark absorption lines”)? explain briefly.

On a day when the barometer reads 75.23 cm, a reaction vessel holds 250 ml of ideal gas at 20 celsius. an oil manometer ( ρ= 810 kg/m^3) reads the pressure in the vessel to be 41 cm of oil and below atmospheric pressure. what volume will the gas occupy under s.t.p.?

Two resistors r1 and r2 may be connected either in series or parallel across an ideal battery with emf ε. we desire the rate of energy dissipation of the parallel combination to be 8.75 times that of the series combination. if r1 = 105 ω, what are the (a) smaller and (b) larger of the two values of r2 that result in that dissipation rate?