In a water pistol, a piston drives water through a large...

In a water pistol, a piston drives water through a large tube of area $A_{1}$ into a smaller tube of area $A_{2}$ as shown in Figure $\mathrm{P} 14.46 .$ The radius of the large tube is $1.00 \mathrm{~cm}$ and that of the small tube is $1.00 \mathrm{~mm}$. The smaller tube is $3.00 \mathrm{~cm}$ above the larger tube. (a) If the pistol is fired horizontally at a height of $1.50 \mathrm{~m},$ determine the time interval required for the water to travel from the nozzle to the ground. Neglect air resistance and assume atmospheric pressure is $1.00 \mathrm{~atm} .$ (b) If the desired range of the stream is $8.00 \mathrm{~m},$ with what speed $v_{2}$ must the stream leave the nozzle? (c) At what speed $v_{1}$ must the plunger be moved to achieve the desired range? (d) What is the pressure at the nozzle? (e) Find the pressure needed in the larger tube. (f) Calculate the force that must be exerted on the trigger to achieve the desired range. (The force that must be exerted is due to pressure over and above atmospheric pressure.)

In a water pistol, a piston drives water through a large tube of area $A_{1}$ into a smaller tube of area $A_{2}$ as shown in Figure $\mathrm{P} 14.46 .$ The radius of the large tube is $1.00 \mathrm{~cm}$ and that of the small tube is $1.00 \mathrm{~mm}$. The smaller tube is $3.00 \mathrm{~cm}$ above
the larger tube.
(a) If the pistol is fired horizontally at a height of $1.50 \mathrm{~m},$ determine the time interval required for the water to travel from the nozzle to the ground. Neglect air resistance and assume atmospheric pressure is $1.00 \mathrm{~atm} .$
(b) If the desired range of the stream is $8.00 \mathrm{~m},$ with what speed $v_{2}$ must the stream leave the nozzle?
(c) At what speed $v_{1}$ must the plunger be moved to achieve the desired range?
(d) What is the pressure at the nozzle?
(e) Find the pressure needed in the larger tube.
(f) Calculate the force that must be exerted on the trigger to achieve the desired range. (The force that must be exerted is due to pressure over and above atmospheric pressure.)

Key Concepts

Projectile Motion

Projectile motion is the study of an object’s movement under the influence of gravity, usually after an initial velocity is imparted. In this context, the water stream leaving the nozzle behaves like a projectile, following a parabolic trajectory. The time it takes for the water to hit the ground is determined only by its vertical displacement and gravitational acceleration, independent of its horizontal velocity.

Continuity Equation

The continuity equation is a fundamental principle in fluid mechanics stating that the mass flow rate remains constant throughout a streamline for an incompressible fluid. This implies that the product of the cross-sectional area and the fluid velocity at any two points along the flow (such as the large and the small tubes of the water pistol) must be equal. It is used to relate the speed of water in the larger tube to the speed in the smaller, nozzle tube.

Bernoulli's Equation

Bernoulli's equation is based on the conservation of energy for flowing fluids. It relates the pressure, kinetic energy per unit volume, and potential energy per unit volume at different points along a streamline. In this problem, Bernoulli's equation connects the pressures in the larger tube, the nozzle, and the change in elevation, thereby allowing calculation of the required pressure that drives the water through the system.

Hydrostatic Pressure

Hydrostatic pressure refers to the change in pressure experienced by a fluid due to differences in height. The vertical separation between the larger tube and the nozzle introduces a gravitational potential energy term that must be overcome by the pressure difference. This is essential for determining the pressure required at various points in the water pistol system, particularly as water moves upwards before being expelled.

Force and Pressure Relationship

The relationship between force and pressure is given by the equation F = P × A, where F is the force, P is the pressure, and A is the area over which the pressure acts. In the context of the water pistol, calculating the force that must be applied on the plunger or trigger requires understanding how the pressure (above atmospheric pressure) translates into a mechanical force over a given area. This connects the fluid dynamics aspect with the mechanical actuation of the device.

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