Question

The velocity of water discharged from a nozzle is given by the formula \(V=12.14 \sqrt{P}\), where \(V\) is the velocity in feet per second and \(P\) is the pressure at the nozzle in pounds per square inch. Find the velocity of water if the nozzle pressure is 64 pounds per square inch.

Short answer

The velocity is 97.12 feet per second.

Step-by-step solution

Identify the Given Values

We start with identifying the values given in the problem. The pressure \( P \) at the nozzle is given as 64 pounds per square inch.

Use the Formula for Velocity

We have a formula for velocity, \( V = 12.14 \sqrt{P} \), where \( V \) is the velocity of the water in feet per second. We'll substitute the given pressure \( P = 64 \) into this formula.

Substitute the Given Pressure into the Formula

Replace \( P \) in the formula with 64: \( V = 12.14 \sqrt{64} \).

Calculate the Square Root of Pressure

Calculate \( \sqrt{64} \). Since 64 is a perfect square, \( \sqrt{64} = 8 \).

Compute the Velocity

Substitute \( 8 \) back into the velocity formula: \( V = 12.14 \times 8 \).

Final Calculation

Perform the multiplication: \( 12.14 \times 8 = 97.12 \).

Conclusion

The velocity of the water discharged from the nozzle is 97.12 feet per second.

Key concepts

Nozzle Pressure

Nozzle pressure is a critical concept in fluid dynamics. It refers to the force exerted by the fluid at the nozzle's exit, typically measured in pounds per square inch (psi). Understanding nozzle pressure is essential for calculating the velocity of fluids like water when discharged from a nozzle. When the pressure at the nozzle increases, the velocity of the water being ejected also increases, due to the relationship defined by specific formulas. This relationship showcases the principle that, under controlled conditions, higher pressure results in greater kinetic energy for the fluid, leading to faster discharge speeds. By plugging in the pressure value into a velocity formula, one can determine how fast the fluid will be moving at the point of exit. In engineering and physics applications, knowing the correct nozzle pressure helps in designing systems that require precise control over fluid flow, such as in irrigation systems, fire hoses, and various industrial applications.

Feet per Second

Velocity, when expressed in feet per second, gives a measure of how fast an object or fluid moves in one second. This unit is very common in physics and engineering, especially when dealing with speeds close to the earth’s surface. To calculate the velocity in feet per second, particularly for our exercise, a formula involving the square root of the nozzle pressure is used. Knowing the nozzle pressure allows one to solve for the discharge velocity effectively. For instance, using the formula \( V = 12.14 \sqrt{P} \), the value you get after calculation shows how far the water travels in one second after exiting the nozzle.This measurement unit is particularly helpful in fields that demand precision over short periods, guiding design and operational decisions especially in fluid transport and dynamic systems.

Physics Formula

The physics formula used in this exercise is a classic example of how mathematical expressions can predict physical behavior. The formula \( V = 12.14 \sqrt{P} \) is designed to calculate the velocity of water coming out of a nozzle based on a given pressure.In this formula:

• \( V \) stands for velocity, which we aim to calculate.
• \( P \) represents the pressure at the nozzle, given in pounds per square inch.
• The constant 12.14 is a conversion factor that aligns the formula to produce results in feet per second when using pressure in psi.

The square root function in the formula reflects the physical reality that forces and pressures in fluid dynamics often translate into velocity in a non-linear way. By substituting the pressure into the formula, it gives an accurate prediction of how fast the fluid exits the nozzle, enabling practical applications in designing systems that rely on precise fluid ejection.