91Ó°ÊÓ

At provides a review of the concepts that are important in this problem. 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 \(0.050 \mathrm{~kg}\) of blood is pushed out of the heart with a velocity of \(+0.25 \mathrm{~m} / \mathrm{s}\) and that the mass of the patient and platform is \(85 \mathrm{~kg}\). Assuming that the patient does not slip with respect to the platform, and that the patient and platform start from rest, determine the recoil velocity.

Step by step solution

01

Understand the Concept of Momentum Conservation

In physics, momentum conservation means that the total momentum of a closed system is constant if no external forces act on it. When the heart pushes blood forward, there is an equal and opposite reaction that causes the platform and patient to recoil backwards.

02

Identify Given Information

We have the following data: the mass of the blood pushed out is \(0.050 \, \text{kg}\); the velocity of the blood is \(+0.25 \, \text{m/s}\); the total mass of the patient and platform is \(85 \, \text{kg}\). Both the initial velocities of the blood and the platform/patient system are \(0 \, \text{m/s}\).

03

Apply the Conservation of Momentum Equation

The initial momentum of the system is zero. According to the conservation of momentum:\[m_{\text{blood}} \cdot v_{\text{blood}} + m_{\text{platform}} \cdot v_{\text{platform}} = 0\]\[0.050 \, \text{kg} \cdot 0.25 \, \text{m/s} + 85 \, \text{kg} \cdot v_{\text{platform}} = 0\]

04

Solve for the Recoil Velocity

Rearranging the equation to solve for \(v_{\text{platform}}\):\[v_{\text{platform}} = -\frac{0.050 \, \text{kg} \cdot 0.25 \, \text{m/s}}{85 \, \text{kg}}\]Calculate \(v_{\text{platform}}\):\[v_{\text{platform}} = -\frac{0.0125 \, \text{kg} \cdot \text{m/s}}{85 \, \text{kg}} \approx -0.000147 \, \text{m/s}\].

05

Consider the Direction of Recoil

The negative sign in \(v_{\text{platform}}\) indicates that the recoil velocity is in the opposite direction to the velocity of the blood, which is expected as per the law of conservation of momentum.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ballistocardiograph

A ballistocardiograph is a medical device used to measure the mechanical activity of the heart. This instrument detects the recoil movements of the body as blood is ejected from the heart with each heartbeat. It's placed on a frictionless platform to measure the body’s subtle movements without resistance interfering. The platform acts as an isolated system where momentum changes can be accurately observed. This setup allows physicians to gather valuable information about cardiac function and helps in diagnosing heart conditions efficiently. By analyzing the recoil, healthcare providers can better understand heart dynamics, like stroke volume and cardiac output.

Recoil Velocity

Recoil velocity refers to the backward movement velocity experienced when an object, like a heart, propels something forward. In this context, when blood is expelled from the heart, the platform and patient's body recoil. The mass and velocity of the blood determine how significant this recoil will be. Essentially, the heart (with the body and platform) moves in the opposite direction of the blood flow due to the conservation of momentum. Understanding recoil velocity is crucial in analyzing systems where one part of a system impacts another through action-reaction pairs, such as in the functioning of a ballistocardiograph.

Momentum Conservation Equation

The momentum conservation equation is a fundamental concept in physics that states momentum in a closed system remains constant. In the scenario given, the momentum of the blood ejected from the heart is balanced by the momentum of the recoiling body and platform. This principle is articulated by the equation:

• Initial total momentum = Final total momentum
• The equation used: \[m_{\text{blood}} \cdot v_{\text{blood}} + m_{\text{platform}} \cdot v_{\text{platform}} = 0\]

This equation allows us to solve for unknown values, such as recoil velocity, by rearranging terms to isolate the desired variable. Mastery of using such equations is essential for solving physics problems that involve momentum exchanges.

Physics Problem Solving

Physics problem solving involves a structured approach to understand and solve complex concepts. It requires identifying known quantities, applying relevant equations, and logically arranging information to find unknowns. Key steps include:

• Understanding the problem statement and identifying all known variables, such as masses and velocities.
• Using appropriate physical laws, like the conservation of momentum, to set up equations.
• Rearranging these equations to solve for the unknown variable, such as the recoil velocity in our example.
• Checking the logic and direction of the solution to ensure it aligns with physical expectations.

This systematic process helps students approach physics tasks methodically, enhancing their problem-solving skills and ensuring accurate solutions.