A ball is dropped from a height \(h\) onto the floor and keeps bouncing. No energy is dissipated, so the ball regains the original height \(h\) after each bounce. Sketch the graph for \(y(t)\) and list several features of the graph that indicate that this motion is not SHM.

Initially the ball is dropped from height \(h\), and accelerates under the influence of gravity. At time \(t=0\), the ball is at a height \(y=h\) and its velocity is zero. As time progresses, the ball falls, its height decreases with increasing velocity due to gravity until it hits the floor. After each bounce, however, the ball regains its original height due to the lack of energy dissipation.

The graph of \(y(t)\) should represent the described motion, consisting of parabolic segments representing free fall and sudden transitions at the points where the ball hits the floor and starts rising again. The sequential parabolic segments should be horizontally mirrored with respect to the bounce points. The bounces themselves will appear as vertical asymptotes in the graph.

Compare the graph of \(y(t)\) with the sinusoidal graph of SHM. Some features that indicate this motion is not SHM are: 1. The graph is not sinusoidal - it consists of segments of parabolas connected with vertical asymptotes. 2. Energy in SHM is distributed between potential and kinetic energy, while in this case, the kinetic energy is quickly converted to potential energy at the bounce points. 3. The motion in SHM has a continuous derivative, whereas the derivative of \(y(t)\) for the bounce motion is discontinuous at the bounce points. 4. The acceleration in SHM is proportional to displacement with a negative constant, but the acceleration during the ball's free fall is constant and not dependent on the displacement. These features confirm that the given motion is not Simple Harmonic Motion.