For tests using a ballistocardiograph, a patient lies on a horizontal platform that is supported on jets of air. Because of the air jets, the friction impeding the horizontal motion of the platform is negligible. Each time the heart beats, blood is pushed out of the heart in a direction that is nearly parallel to the platform. Since momentum must be conserved, the body and the platform recoil, and this recoil can be detected to provide information about the heart. For each beat, suppose that 0.050 kg of blood is pushed out of the heart with a velocity of +0.35 m/s and that the mass of the patient and the platform is 75 kg. Assuming that the patient does not slip with respect to the platform, and that the patient and the platform start from rest, determine the recoil velocity
To determine the recoil velocity of the patient and the platform, we can use the principle of conservation of momentum. The total initial momentum of the system (patient + platform) is zero because both start from rest. After the blood is pushed out, the total final momentum of the system will also be zero, as there is no external force acting on the system.
The momentum before and after the blood is pushed out can be calculated using the equation:
Initial momentum = Final momentum
The initial momentum of the system is given by the mass of the blood (0.050 kg) multiplied by its velocity (+0.35 m/s):
Initial momentum = (0.050 kg) * (+0.35 m/s) = +0.0175 kg·m/s
Since the patient and the platform are initially at rest, their initial momentum is zero. Therefore, the final momentum of the patient and the platform must also be -0.0175 kg·m/s to maintain the conservation of momentum.
The final momentum of the system can be calculated using the equation:
Final momentum = (mass of patient + mass of platform) * (recoil velocity)
Final momentum = (75 kg + mass of platform) * (recoil velocity)
Setting the final momentum equal to the initial momentum and solving for the recoil velocity:
(75 kg + mass of platform) * (recoil velocity) = -0.0175 kg·m/s
Now we need to find the mass of the platform. It is not provided in the question, so we cannot determine the exact recoil velocity without that information.