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One way to administer an inoculation is with a 鈥済un鈥� that shoots the vaccine through a narrow opening. No needle is necessary, for the vaccine emerges with sufficient speed to pass directly into the tissue beneath the skin. The speed is high, because the vaccine \(\left(\rho=1100 \mathrm{kg} / \mathrm{m}^{3}\right)\) is held in a reservoir where a high pressure pushes it out. The pressure on the surface of the vaccine in one gun is 4.1 106 Pa above the atmospheric pressure outside the narrow opening. The dosage is small enough that the vaccine鈥檚 surface in the reservoir is nearly stationary during an inoculation. The vertical height between the vaccine鈥檚 surface in the reservoir and the opening can be ignored. Find the speed at which the vaccine emerges.

Step by step solution

01

Understand the basics of Bernoulli's Equation

Bernoulli's equation states that for an incompressible, non-viscous fluid, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant along any streamline. The equation is given by: \[ P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 \]where:- \( P \) is the fluid pressure,- \( \rho \) is the fluid density,- \( v \) is the fluid velocity,- \( g \) is the acceleration due to gravity (which can be ignored as per the problem statement), and- \( h \) is the height above a reference point.

02

Simplify Bernoulli's Equation for this situation

Since the vertical height difference is negligible and we assume the reservoir's surface velocity \(v_1\) is nearly zero, we can simplify Bernoulli鈥檚 equation to:\[ P_1 = P_2 + \frac{1}{2} \rho v_2^2 \]Rearranging for \(v_2\), we have:\[ v_2 = \sqrt{\frac{2(P_1 - P_2)}{\rho}} \]

03

Identify the pressures

Here, \(P_1\) is the pressure in the reservoir, and \(P_2\) is atmospheric pressure. The problem states that the pressure difference \(P_1 - P_2 = 4.1 \times 10^6 \) Pa.

04

Calculate the exit speed

Plug in the given values, with \( \rho = 1100 \; \mathrm{kg/m}^3 \):\[ v_2 = \sqrt{\frac{2 \times 4.1 \times 10^6}{1100}} \]Calculate:\[ v_2 = \sqrt{\frac{8.2 \times 10^6}{1100}} \approx \sqrt{7454.545} \approx 86.37 \; \mathrm{m/s} \]

05

Present the final answer

The speed at which the vaccine emerges from the gun is approximately 86.37 m/s.

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Key Concepts

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

Fluid Dynamics

Fluid dynamics is a branch of physics that studies the motion of fluids鈥攍iquids and gases. One of the most significant principles in fluid dynamics is Bernoulli's Equation. It provides a way to understand how fluid flows behave under various conditions. In the context of vaccines delivered through a jet or gun, fluid dynamics helps us predict the speed and pressure changes of the fluid as it moves from a high-pressure reservoir to the atmosphere.

• Flows in fluid dynamics can be steady or unsteady. Steady flow means that at any given point, the fluid's velocity doesn't change over time.
• This field analyzes various flow paths called streamlines, which represent the trajectories that particles of a fluid would follow.

Comprehending fluid dynamics helps in designing effective delivery systems for vaccines that require high-speed injections without needles.

Pressure Energy

Pressure energy in a fluid is related to the force the fluid exerts per unit area due to its internal energy. In Bernoulli's Equation, pressure energy is one of the key factors to consider. It describes how fluid pressure can convert to kinetic energy as fluid moves through different points.

• It's measured in pascals (P) and is a form of potential energy within the fluid.
• In the context of the vaccine injection, a high pressure of 4.1 x 10⁶ Pa is placed on the vaccine in the reservoir, which supplies the force needed for the vaccine to exit at high speed.

Understanding pressure energy is crucial for designing systems that maintain the right pressure level to ensure the fluid exits at the desired speed.

Kinetic Energy

Kinetic energy refers to the energy a fluid possesses due to its motion. In Bernoulli's equation, it is represented as the term \( \frac{1}{2} \rho v^2 \), where \( \rho \) is the density, and \( v \) is the velocity of the fluid. When a fluid accelerates or decelerates, its kinetic energy changes.

• When liquid emerges from a nozzle, kinetic energy is maximized at the point of exit because the velocity is highest there.
• For the vaccine gun, as the vaccine speeds up while passing through the narrow opening, the kinetic energy increases until the maximum exit velocity of 鈮� 86.37 m/s is reached.

Kinetic energy is an essential part of understanding how fluids interact with their surroundings as their speed changes.

Potential Energy

Potential energy in fluids is generally related to their position in a gravitational field and is expressed as \( \rho g h \) in Bernoulli's equation. However, in situations like the vaccine delivery system mentioned, the potential energy due to height differences is negligible and often ignored.

• Potential energy becomes significant when fluid moves between different vertical heights.
• Since the difference in height is minor in the vaccine scenario, the focus is more on pressure and kinetic energies.

While potential energy is a vital concept in many fluid situations, in this case, it takes a backseat to the overwhelming effects of pressure and kinetic energy changes.