The Conservation of Mass and Momentum

Oct 13, 2025 by miodeya.com 102 views

Introduction

In this article, we will explore the concept of conservation of mass and momentum in a garden hose scenario. We will use the given information to calculate the new velocity of the water after Dasha puts her thumb over the end of the hose, reducing its area.

The Conservation of Mass and Momentum

The conservation of mass and momentum is a fundamental principle in physics that states that the total mass and momentum of a closed system remain constant over time. In the context of a garden hose, this means that the mass of the water flowing out of the hose is equal to the mass of the water flowing into the hose.

The Continuity Equation

The continuity equation is a mathematical expression of the conservation of mass. It states that the mass flow rate of a fluid is equal to the product of the fluid's density, velocity, and cross-sectional area.

\rho \cdot A \cdot v = \text{constant}

where $\rho$ is the density of the fluid, $A$ is the cross-sectional area of the hose, and $v$ is the velocity of the fluid.

The Given Information

We are given the following information:

The radius of the garden hose is $0.0120 \, \text{m}$ .
The initial velocity of the water is $2.88 \, \text{m/s}$ .
The area of the hose is reduced to $1.05 \times 10^{-4} \, \text{m}^2$ after Dasha puts her thumb over the end.

Calculating the Initial Mass Flow Rate

To calculate the initial mass flow rate, we need to calculate the initial velocity of the water. We can do this using the continuity equation.

\rho \cdot A \cdot v = \text{constant}

We know that the density of water is approximately $1000 \, \text{kg/m}^3$ . We can plug in the values for the radius and area of the hose to get:

1000 \, \text{kg/m}^3 \cdot \pi \cdot (0.0120 \, \text{m})^2 \cdot v = \text{constant}

Simplifying the equation, we get:

1000 \, \text{kg/m}^3 \cdot 1.44 \times 10^{-4} \, \text{m}^2 \cdot v = \text{constant}

1.44 \, \text{kg/s} \cdot v = \text{constant}

Now, we can plug in the value for the initial velocity to get:

1.44 \, \text{kg/s} \cdot 2.88 \, \text{m/s} = 4.15 \, \text{kg/m}^2\text{s}

Calculating the New Velocity

Now that we have the initial mass flow rate, we can calculate the new velocity of the water after Dasha puts her thumb over the end of the hose. We can do this by using the continuity equation again.

\rho \cdot A \cdot v = \text{constant}

We know that the density of water is approximately $1000 \, \text{kg/m}^3$ . We can plug in the values for the area of the hose and the initial mass flow rate to get:

1000 \, \text{kg/m}^3 \cdot 1.05 \times 10^{-4} \, \text{m}^2 \cdot v = 4.15 \, \text{kg/m}^2\text{s}

Simplifying the equation, we get:

1.05 \, \text{kg/s} \cdot v = 4.15 \, \text{kg/m}^2\text{s}

Now, we can solve for the new velocity:

v = \frac{4.15 \, \text{kg/m}^2\text{s}}{1.05 \, \text{kg/s}}

v = 3.95 \, \text{m/s}

Conclusion

In this article, we used the continuity equation to calculate the new velocity of the water after Dasha puts her thumb over the end of the hose. We found that the new velocity is approximately $3.95 \, \text{m/s}$ .

References

Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

Discussion

What do you think about the concept of conservation of mass and momentum? How does it apply to real-world scenarios? Share your thoughts in the comments below!

The Continuity Equation
The Conservation of Mass and Momentum
Fluid Dynamics
A Garden Hose and the Conservation of Mass and Momentum: Q&A ===========================================================

Introduction

In our previous article, we explored the concept of conservation of mass and momentum in a garden hose scenario. We used the continuity equation to calculate the new velocity of the water after Dasha puts her thumb over the end of the hose. In this article, we will answer some frequently asked questions about the concept of conservation of mass and momentum.

Q: What is the continuity equation?

A: The continuity equation is a mathematical expression of the conservation of mass. It states that the mass flow rate of a fluid is equal to the product of the fluid's density, velocity, and cross-sectional area.

Q: What is the significance of the continuity equation?

A: The continuity equation is significant because it helps us understand how the mass of a fluid changes as it flows through a pipe or hose. It is a fundamental principle in fluid dynamics and is used to calculate the velocity of a fluid in a pipe or hose.

Q: How does the continuity equation apply to real-world scenarios?

A: The continuity equation applies to many real-world scenarios, such as:

Water flowing through a pipe or hose
Air flowing through a duct or pipe
Gas flowing through a pipe or hose
Blood flowing through the circulatory system

Q: What is the difference between the continuity equation and the conservation of mass and momentum?

A: The continuity equation is a mathematical expression of the conservation of mass, while the conservation of mass and momentum is a fundamental principle that states that the total mass and momentum of a closed system remain constant over time.

Q: Can the continuity equation be used to calculate the velocity of a fluid in a pipe or hose?

A: Yes, the continuity equation can be used to calculate the velocity of a fluid in a pipe or hose. We can plug in the values for the density of the fluid, the cross-sectional area of the pipe or hose, and the mass flow rate to get the velocity.

Q: What are some common applications of the continuity equation?

A: Some common applications of the continuity equation include:

Calculating the velocity of water flowing through a pipe or hose
Calculating the flow rate of a fluid in a pipe or hose
Calculating the pressure drop across a pipe or hose
Calculating the energy losses in a pipe or hose

Q: Can the continuity equation be used to calculate the pressure drop across a pipe or hose?

A: Yes, the continuity equation can be used to calculate the pressure drop across a pipe or hose. We can use the equation:

\Delta P = \frac{\rho \cdot v^2}{2}

where $\Delta P$ is the pressure drop, $\rho$ is the density of the fluid, and $v$ is the velocity of the fluid.

Q: What are some common mistakes to avoid when using the continuity equation?

A: Some common mistakes to avoid when using the continuity equation include:

Not accounting for the density of the fluid
Not accounting for the cross-sectional area of the pipe or hose
Not accounting for the mass flow rate
Not using the correct units for the variables

Conclusion

In this article, we answered some frequently asked questions about the concept of conservation of mass and momentum. We hope that this article has been helpful in understanding the continuity equation and its applications.

References

Halliday, D., Resnick, R., & Walker, J. (2013). Fundamentals of physics. John Wiley & Sons.
Serway, R. A., & Jewett, J. W. (2018). Physics for scientists and engineers. Cengage Learning.

Discussion

What do you think about the concept of conservation of mass and momentum? How does it apply to real-world scenarios? Share your thoughts in the comments below!