In research in cardiology and exercise physiology, it is often important to know

Question

In research in cardiology and exercise physiology, it is often important to know the mass of blood pumped by a person's heart in one stroke. This information can be obtained by means of a ballistocardiograph. The instrument works as follows: The subject lies on a horizontal pallet floating on a film of air. Friction on the pallet is negligible. Initially, the momentum of the system is zero. When the heart beats, it expels a mass m of blood into the aorta with speed v, and the body and platform move in the opposite direction with speed V. The speed of the blood can be determined independently (for example, by observing an ultrasound Doppler shift). Assume that the blood's speed is 49.5 cm/s in one typical trial. The mass of the subject plus the pallet is 54.0 kg. The pallet moves 5.70 × 10-5 m in 0.160 s after one heartbeat. Calculate the mass of blood that leaves the heart. Assume that the mass of blood is negligible compared with the total mass of the person. This simplified example illustrates the principle of ballistocardiography, but in practice a more sophisticated model of heart function is used.

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Explanation / Answer

This is a basic conservation of momentum problem. It simply means that the total momentum remains unchanged unless an external force acts on the system (in this problem, no such force acts on the system).

Recall that momentum is simply (mass)*(velocity).

Thus, we can write this problem as follows:

(mass of blood) * (velocity of blood) = (combined mass of subject and pallet) * (velocity of subject and pallet)

Remember that velocity is meters/second, and we're given the distance the pallet moves and the time it takes to move that far.

To keep units consistent, let's recast 49.5cm/s as 0.495 m/s.

Or, to put in the numbers you have:

(mass of blood)*(0.495m/s)=(54.0kg)*(5.70*10^-5/0.160... m/s)

Now, let's solve for the mass of the blood by dividing both sides by 0.500m/s.

This results in the following:

(mass of blood) = [(54.0kg)*(5.70*10^-5/0.160 m/s)]/(0.495m/s)

The velocity units (m/s) cancel out, so we're left with 0.0388kg

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