Final answer:
The velocity of water squirting from the needle of a syringe can be calculated using Bernoulli's equation and the conservation of flow rate by considering the force applied, and the cross-sectional areas of both the barrel and the needle.
Step-by-step explanation:
To calculate the velocity of water as it squirts from the needle of a syringe when a force is applied, we can use the principle of conservation of flow rate and apply Bernoulli's equation. For a syringe, the conservation of flow rate tells us that the product of cross-sectional area and velocity is the same at any two points along the flow. Since the areas of the barrel and the needle are different, the velocities will also be different.
To find the velocity, we first calculate the area of the needle's opening (Aneedle) and the area of the barrel's cross-section (Abarrel). The force applied to the plunger (F) creates pressure that pushes the water through the syringe. Using Aneedle = π * (Dneedle/2)2 and Abarrel = π * (Dbarrel/2)2, and the relation F = Abarrel * P, where P is the pressure, we can derive the velocity of water exiting the needle (Vneedle) by Vneedle = Abarrel * Vbarrel / Aneedle.
The velocity of the barrel can be assumed to be negligible due to the large cross-sectional area compared to the needle, thus Vbarrel is practically zero, and the velocity Vneedle can be primarily determined by the pressure created by the applied force.