Question

Allison throws a tennis ball at time t = 0 from a height h above the ground equal to her own height. We assume that the positive horizontal direction is the positive x-direction, and the positive vertical direction is the positive y-direction. The ball has a positive initial horizontal velocity vx, and no initial vertical velocity. At times t1 and t2, the ball bounces on the ground for the first, and second times respectively after being thrown. We assume that each bounce is instantaneous. We also assume that each time the ball bounces, its horizontal speed just after the bounce is the same as its horizontal speed just before the bounce, but itsvertical speed just after the bounce is 3/4 its vertical speed just before the bounce.

2.1. Let x, y, vx, vy, ax, ay be the x- and y-components of the ball’s position, velocity, and acceleration respectively. Sketch graphs of the following quantities between time t = 0 and time t = t2. You may ignore what the graph looks like at the times t1 and t2 when drawing your graphs.

y versus x x versus t y versus t vx versus t vy versus t ax versus t ay versus t

Your sketches do not need to contain labels indicating values on the axes – they should, however, have the correct qualitative features, like the shape of the graph, and relative size of various parts of the graph.

2.2. Redraw the sketch of y versus x. At each of the points on your sketch corresponding to the bounce times t1 and t2 draw what the velocity vector of the ball looks like just before the bounce and just after the bounce. Between times t1 and t2, draw what the velocity vector looks like at three different points with one of the points before the ball reaches its apex, one point being its apex, and one point after it reaches its apex. Your velocity vectors should have the appropriate directions and roughly appropriate lengths relative to one another.

2.3. Redraw the sketch of y versus x once again, but this time draw what the ball’s acceleration vector looks like between times t1 and t2 at three points: one before the apex, one at the apex, and one after. Your acceleration vectors should have the appropriate directions and roughly appropriate lengths relative to one another.

Answer 1