Calculate the relativistic energy e and momentum p of a proton of mass m = 1 gev/c^2 moving at velocity u = 0.9c in the laboratory frame. then calculate the " in-variant, mass" k^2-e^2-p^2c^2 in the laboratory frame. compare your result to the invariant mass k'^2 in the frame where the proton is at rest (v'=0) what do you conclude?

Adiver named jacques observes a bubble of air rising from the bottom of a lake (where the absolute pressure is 3.50 atm) to the surface (where the pressure is 1.00 atm). the temperature at the bottom is 4.00 ∘c, and the temperature at the surface is 23.0 ∘c.what is the ratio of the volume of the bubble as it reaches the surface (vs) to its volume at the bottom (vb)? if jaques were to hold his breath the air in his lungs would be kept at a constant temperature. would it be safe for jacques to hold his breath while ascending from the bottom of the lake to the surface?

In a game of pool, ball a is moving with a velocity v0 of magnitude v0 = 15 ft/s when it strikes balls b and c, which are at rest and aligned as shown. knowing that after the collision the three balls move in the directions indicated and assuming frictionless surfaces and perfectly elastic impact (that is, conservation of energy), determine the magnitudes of the velocities va, vb, and vc.

Relative to the ground below, how many joules of potential energy does a 1000-kg boulder have at the top of a 5-m ledge?