A differential equation for the velocity of a falling mass m subjected to air resistance proportional to the square of the instantaneous velocity is m = m − 2 where > 0 is a constant of proportionality. The positive direction is downward. (a) Solve the equation subject to the initial condition (0) = 0. Hint: ∫ 2−2 = 1 tanh−1 ( ) + c (b)Use the solution in part (a) to determine the limiting, or terminal, velocity of the mass. That is, what happens to () as → [infinity]?

Ametal ring 4.20 cm in diameter is placed between the north and south poles of large magnets with the plane of its area perpendicular to the magnetic field. these magnets produce an initial uniform field of 1.12 t between them but are gradually pulled apart, causing this field to remain uniform but decrease steadily at 0.240 t/s . (a) what is the magnitude of the electric field induced in the ring? (b) in which direction (clockwise or counterclockwise) does the current flow as viewed by someone on the south pole of the magnet?