Question

A graph of acceleration of the table versus time, termed a ballistocardiogram, is generated. Based on these measurements, the acceleration of the blood ejected by the heart can be determined. Patients with low blood accelerations generally have weakened heart muscles.
A sketch of a single cycle of a ballistocardiogram is given in the figure. (Figure 1) . The units of the graph are arbitrary and linear for both time,t, and acceleration, a.At what time (in the arbitrary time units of the graph) is the speed of the table (and hence the speed of the blood in the opposite direction) a maximum?

Answer 1

Answer on Question #71712-Physics-Other

A graph of acceleration of the table versus time, termed a ballistocardiogram, is generated. Based on these measurements, the acceleration of the blood ejected by the heart can be determined. Patients with low blood accelerations generally have weakened heart muscles.

A sketch of a single cycle of a ballistocardiogram is given in the figure. (Figure 1). The units of the graph are arbitrary and linear for both time, t, and acceleration, a. At what time (in the arbitrary time units of the graph) is the speed of the table (and hence the speed of the blood in the opposite direction) a maximum?

Figure 1: acceleration-time graph

Solution

The speed can have a maximum when acceleration is zero: points B and D.

The second derivative test for a function is,

$$\frac {d ^ {2} v}{d t ^ {2}} < 0$$ $$\frac {d a}{d t} < 0$$

Thus, the slope of the acceleration should be negative. It is point B. So,

$$t = 3.$$

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