Consider the following wave functions (where a, b, k, and a are constants): (i) ψ(x) = a sin(kx) (iv) ψ(x) = ae-ikx (i) ψ(x) = b cos(kx) (v) ψ(x) = be+ikx (iii) ψ(x) = a/x (a) determine which of the above func- (vi) ψ(x) be-ax tions are eigenfunctions of the momentum operator, px, and identify the correspond- ing eigenvalue. (b) determine which of the above functions are eigenfunctions of the kinetic energy oper- ator, kx, and determine the corresponding eigenvalue. 4. evaluate the momentum expectation values of the free particle wavefunctions ex and etilx (where k is a positive constant). what do your results tell you about the direction in which free particles described by each of these wavefunctions are moving?

Which best represents the law of conservation of mass ?

Alfred pushes on a heavy box, but cannot move it. the box has a lot of inertia motion friction gravity

Suppose water is leaking from a tank through a circular hole of area ah at its bottom. when water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cah 2gh , where c (0 < c < 1) is an empirical constant. a tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (assume the removed apex of the cone is of negligible height and volume.) (a) suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. the differential equation governing the height h in feet of water leaking from a tank after t seconds is dh dt = − 5 6h3/2 . if the height of the water is initially 8 feet, how long will it take the tank to empty? (round your answer to two decimal places.)