A hypodermic syringe contains a medicine having the density...

A hypodermic syringe contains a medicine having the density of water (Fig. P14.49). The barrel of the syringe has a cross-sectional area $A=2.50 \times 10^{-5} \mathrm{m}^{2},$ and the needle has a cross-sectional area $a=1.00 \times 10^{-8} \mathrm{m}^{2} .$ Int he absence of a force on the plunger, the pressure everywhere is 1 atm. A force $\overrightarrow{\mathbf{F}}$ of magnitude 2.00 $\mathrm{N}$ acts on the plunger, making medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip.

Key Concepts

Continuity Equation

The continuity equation reflects the conservation of mass in fluid dynamics. For an incompressible fluid, the product of the cross-sectional area and the velocity must remain constant along the flow path. This concept ensures that even when a fluid passes through regions of differing cross-sectional areas, such as from a syringe barrel to a needle, the flow rate remains constant.

Fluid Density

Fluid density is the mass per unit volume of a fluid and is a critical parameter when relating pressure differences to fluid acceleration. In applying Bernoulli's Principle, the density of the fluid is used to calculate the conversion of pressure energy into kinetic energy, thereby determining the speed of the fluid as it exits.

Pressure

Pressure is defined as the force applied per unit area and is a fundamental concept in fluid mechanics. In practical applications, such as in a syringe, applying a force over a specific area increases the pressure in the fluid, which in turn causes the fluid to move from regions of higher pressure to regions of lower pressure.

Bernoulli's Principle

Bernoulli's Principle states that for an incompressible, non-viscous fluid in steady flow, the total mechanical energy (which includes pressure energy, kinetic energy, and potential energy) is conserved along a streamline. This principle is used to relate the pressure difference created by an applied force to the velocity of the fluid as it exits an opening.

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