Halliday, Resnick & Walker — Motion Along a Straight Line: Problem 108

Remote fall detection. Falling is a chronic danger to the elderly and people subject to seizure. Researchers search for ways to detect a fall remotely so that a caretaker can go to the victim quickly. One way is to use a computer system that analyzes the motions of someone on CCTV in real time. The system monitors the vertical velocity of someone and then calculates the vertical acceleration when that velocity changes. If the system detects a large negative (downward) acceleration followed by a briefer positive acceleration and accompanied by a sound burst for the onset of the positive acceleration, a signal is sent to a caretaker. Figure 2.39 gives an idealized graph of vertical velocity \(v\) versus time \(t\) as determined by the system: \(t_1 = 1.0 \text{ s}, t_2 = 2.5 \text{ s}, t_3 = 3.0 \text{ s}, t_4 = 4.0 \text{ s}, v_1 = -7.0 \text{ m/s}\). (The plot on a more realistic graph would be curved.) What are (a) the acceleration during the descent and (b) the upward acceleration during the impact with the floor?