Aheavy ball swings on a string in a circular arc of radius 1 m. q 1 m q ′ p 24 ◦ the two highest points of the ball’s trajectory are q and q ′ ; at these points the string is ±24◦ from the vertical. point p is the lowest point of the ball’s trajectory where the string hangs vertically down. the acceleration of gravity is 9.8 m/s 2 . what is the ball’s speed at the point p? neglect air resistance and other frictional forces. answer in units of m/s. what is the magnitude of the ball’s acceleration at the point p? answer in units of m/s 2 .

Let v1, v2, w be three linearly independent vectors in r 3 . that is, they do not all lie on the same plane. for each of the following (infinite) set of vectors, carefully sketch it in r 3 , and determine whether or not it is a vector space (i.e., a subspace of r 3 ). explain your reasoning

A23 kg log of wood begins from rest, 300 m up a sluice (a water track used to transport logs, think of it as an inclined plane with negligible friction) inclined at 20° to the horizontal. after it reaches the flat waterway at the bottom it collides elastically with a 100 kg block of wood initially at rest near the base of the incline. (a) how long does it take the 23 kg log to travel down the incline? (b) what is the speed of the 23 kg log at the bottom of the incline? (c) what are the velocities of both blocks of wood after the collision? (d) what is the total kinetic energy before the collision? (e) what is the kinetic energy of the 23 kg log after the collision? (f) what is the kinetic energy of the 100 kg block of wood after the collision? (g) what is the total kinetic energy after the collision? (h) compare the total initial and total final kinetic energies. is this consistent with what you would expect for elastic collisions? explain!

Which type of wave can travel without a medium