Q92PP - University Physics with Modern Physics

Question

The velocity of blood in the aorta can be measured directly with ultrasound techniques. A typical graph of blood velocity versus time during a single heartbeat is shown in Fig. P2.92. Which statement is the best interpretation of this graph? (a) The blood flow changes direction at about 0.25 s; (b) the speed of the blood flow begins to decrease at about 0.10 s; (c) the acceleration of the blood is greatest in magnitude at about 0.25 s; (d) the acceleration of the blood is greatest in magnitude at about 0.10 s.

Step-by-Step Solution

Answer

The acceleration of the blood is greater in magnitude at about $0.10\mathrm{s}$.

Step 1: Identification of given data

The time intervals at which blood velocity is shown is,$0.10\mathrm{s},0.25\mathrm{s},0.50\mathrm{s},0.75\mathrm{s},1.00\mathrm{s}$.

The given blood velocity is,$0.2\mathrm{m}/\mathrm{s},0.4\mathrm{m}/\mathrm{s},0.6\mathrm{m}/\mathrm{s},0.8\mathrm{m}/\mathrm{s},1.0\mathrm{m}/\mathrm{s}$

Step 2: Concept of instantaneous acceleration

The ratio of velocity changes during a certain time period to zero.

The instantaneous acceleration is given by,

$a=\frac{d}{dt}\left(\frac{dx}{dt}\right)=\frac{d^{2}x}{d{t}^2}...\left(\mathrm{i}\right)$

Step 3: Evaluate the velocity-time graph of blood in aorta

The direction of blood flow does not alter. The rate of blood flow decreases.

After $0.2\mathrm{s}$, the slope flattens off and blood flow speeds up the most. 

The slope of graph is steeper about$0.10\mathrm{s}$ ,where acceleration of blood is greatest.

Thus, the acceleration of the blood is greatest about $0.10\mathrm{s}$.