Problem 31 - Algebra and Trigonometry

Question

A Jet of Water The power \(P\) of a jet of water is jointly proportional to the cross-sectional area \(A\) of the jet and to the cube of the velocity \(v\) . If the velocity is doubled and the cross-sectional area is halved, by what factor will the power increase?

Step-by-Step Solution

Answer

The power increases by a factor of 4.

Step 1: Understand the Relation

The power of the jet of water, \( P \), is given by an equation that is jointly proportional to the cross-sectional area \( A \) and the cube of the velocity \( v \). This means:\[P = k A v^3\]where \( k \) is a constant of proportionality.

Step 2: Initial Power Expression

The initial power of the jet of water is written with the initial area \( A \) and velocity \( v \):\[P_1 = k A v^3\]

Step 3: Modified Power Expression

When the velocity is doubled (\( 2v \)) and the cross-sectional area is halved (\( \frac{A}{2} \)), the new power \( P_2 \) can be expressed as:\[P_2 = k \left(\frac{A}{2}\right) (2v)^3\]

Step 4: Calculate New Power

Calculate the new power:- The new area is \( \frac{A}{2} \).- The new velocity is \( 2v \), hence its cube is \((2v)^3 = 8v^3\).Plugging these in:\[P_2 = k \frac{A}{2} \cdot 8v^3 = k \cdot 4A \cdot v^3\]

Step 5: Determine the Factor Increase

Compare the new power \( P_2 \) to the initial power \( P_1 \):- Initial power was \( P_1 = k A v^3 \).- New power is \( P_2 = 4 k A v^3 \).Thus, the power increases by a factor of 4.

Key Concepts

Joint Variation

Joint variation occurs when one quantity is proportional to the product of two or more other quantities. In our exercise, we see this concept applied in the relationship between the power of a jet of water and its cross-sectional area and velocity. This relationship is expressed mathematically as:

• Power, \( P \), is jointly proportional to cross-sectional area, \( A \), and the cube of velocity, \( v \).
• This relationship is expressed as \( P = kAv^3 \), where \( k \) is the constant of proportionality.

Joint variation is powerful because it allows us to understand how a change in one or more of the associated variables affects the main variable, here being power.

Cross-Sectional Area

The concept of cross-sectional area refers to the area of a cut through a three-dimensional object, perpendicular to one of its dimensions, at a particular point. In this context, the cross-sectional area \( A \) represents the size of the opening from which the water jet emerges.

• A larger cross-sectional area will allow more water to pass through, increasing the power of the jet.
• Conversely, reducing the area, as in this problem where it is halved, will result in less power if all other factors remain the same.

Understanding how cross-sectional area impacts power is key to visualizing practical applications like hose nozzles or fire hydrants.

Velocity Cube

Velocity is the speed of the water in a given direction. In this exercise, the power of the water jet is influenced by the cube of the velocity \( v^3 \).

• Cubing the velocity implies that small changes in velocity lead to large changes in power since the velocity is expressed as \( v^3 \).
• For example, doubling the velocity from \( v \) to \( 2v \) results in multiplying the original velocity cube by 8, because \( (2v)^3 = 8v^3 \).
• This exponential relationship highlights why increasing velocity has such a significant effect on the power of fluids in motion.

Grasping this concept is crucial for understanding scenarios in fluid dynamics where velocity changes.

Power Increase Factor

The power increase factor quantifies how much the power changes due to alterations in the influencing variables. Here, changes in both velocity and cross-sectional area affect the jet's power.

• The initial setup gives the power as \( k A v^3 \).
• With velocity doubled to \( 2v \) and area halved to \( \frac{A}{2} \), the new power becomes \( 4kAv^3 \).
• This results in a fourfold increase in power, an increase factor of 4, illustrating the combined effect of the adjusted variables.

Calculating power increase factors is valuable for tasks like optimizing water jets or assessing energy expenditure in various engineering applications.