1. The flow rate in a 7-in diameter pipe is 3.2 ft³/s. What is the average velocity in the pipe? 

2. A pipe delivers 239 gal of water a minute. If the velocity of the water is 27 ft/s, what is the diameter of the pipe?

3. Water flows in a pipe of 23-cm diameter with an average velocity of 0.73 m/s. The diameter of the pipe is reduced and the average velocity of the water increases to 1.66 m/s. What is the diameter of the smaller pipe? What is the flow rate?

4. Water in a 5.5-in diameter pipe has a velocity of 97 gal/s; the pipe splits in two to feed two systems. If after splitting one pipe has 3.2-in diameter and the other 1.8-in diameter, what is the flow rate from each pipe?

5. A copper sphere is dropped from a building 273 ft tall. What will be its velocity on impact with the ground? Ignore air resistance.

Let's go step-by-step through each question:

### 1. **Flow rate in a 7-inch diameter pipe with a flow rate of 3.2 ft³/s:**
We need to find the **average velocity** of water in the pipe. The formula to calculate velocity based on flow rate is:

\[
V = \frac{Q}{A}
\]

Where:
- \( V \) is the average velocity (ft/s),
- \( Q \) is the flow rate (ft³/s),
- \( A \) is the cross-sectional area of the pipe (ft²).

For a circular pipe:
\[
A = \pi \left(\frac{D}{2}\right)^2
\]

- Convert the diameter to feet: \( D = \frac{7}{12} \) ft.
- Then, calculate the area \( A \):
\[
A = \pi \left(\frac{7}{12} \times \frac{1}{2}\right)^2 = \pi \left(\frac{7}{24}\right)^2
\]

Once you calculate the area, use the formula for velocity.

---

### 2. **Pipe delivering 239 gallons/minute with a velocity of 27 ft/s:**
We need to find the **diameter** of the pipe. Use the same flow rate formula:

\[
Q = V \times A
\]

Where:
- \( Q \) is the flow rate (ft³/s). First, convert 239 gal/min to ft³/s:
\[
239 \, \text{gal/min} = \frac{239}{7.481} \, \text{ft}^3/\text{min} \approx 31.94 \, \text{ft}^3/\text{min}
\]
\[
Q = \frac{31.94}{60} \, \text{ft}^3/\text{s} \approx 0.532 \, \text{ft}^3/\text{s}
\]
Now, solve for the area \( A \):
\[
A = \frac{Q}{V} = \frac{0.532}{27} \approx 0.0197 \, \text{ft}^2
\]
Using the formula for the area of a circle, solve for the diameter \( D \):
\[
A = \pi \left(\frac{D}{2}\right)^2 \Rightarrow D = 2 \times \sqrt{\frac{A}{\pi}}
\]

---

### 3. **Pipe with water flow changing velocities:**
The **flow rate** remains constant through the pipe. The formula for flow rate is:

\[
Q = A_1 \cdot V_1 = A_2 \cdot V_2
\]

Where:
- \( A_1 \) is the area of the larger pipe,
- \( V_1 = 0.73 \, \text{m/s} \) is the velocity in the larger pipe,
- \( A_2 \) is the area of the smaller pipe,
- \( V_2 = 1.66 \, \text{m/s} \) is the velocity in the smaller pipe.

First, calculate the flow rate \( Q \) using the area of the larger pipe (with diameter \( 23 \, \text{cm} \)):
\[
A_1 = \pi \left(\frac{23}{2 \times 100}\right)^2 \, \text{m}^2
\]
Then solve for \( Q \):
\[
Q = A_1 \cdot V_1
\]
Now, use the same flow rate \( Q \) to find the area of the smaller pipe:
\[
Q = A_2 \cdot V_2
\]
Finally, solve for the diameter of the smaller pipe.

---

### 4. **Flow rate from two split pipes:**
The **total flow rate** before the split is 97 gallons/s. Convert it to cubic feet per second:

\[
97 \, \text{gal/s} = \frac{97}{7.481} \, \text{ft}^3/\text{s} \approx 12.97 \, \text{ft}^3/\text{s}
\]

Now, for each pipe, use the formula for flow rate:

\[
Q_1 = A_1 \cdot V_1 \quad \text{and} \quad Q_2 = A_2 \cdot V_2
\]

Where:
- \( A_1 \) is the area of the 3.2-inch diameter pipe,
- \( A_2 \) is the area of the 1.8-inch diameter pipe,
- The velocities in each pipe depend on the area.

Use the continuity equation:
\[
Q = Q_1 + Q_2
\]
Solve for \( Q_1 \) and \( Q_2 \).

---

### 5. **Velocity of copper sphere dropped from 273 ft:**
This is a **free-fall problem** where we neglect air resistance. The formula for velocity after falling a distance \( h \) is:

\[
v = \sqrt{2gh}


1. The flow rate in a 7-in diameter pipe is 3.2 ft³/s. What is the average velocity in the pipe?
2. A pipe delivers 239 gal of water a minute. If the velocity of the water is 27 ft/s, what is the diameter of the pipe?
3. Water flows in a pipe of 23-cm diameter with an average velocity of 0.73 m/s. The diameter of the pipe is reduced and the average velocity of the water increases to 1.66 m/s. What is the diameter of the smaller pipe? What is the flow rate?
4. Water in a 5.5-in diameter pipe has a velocity of 97 gal/s; the pipe splits in two to feed two systems. If after splitting one pipe has 3.2-in diameter and the other 1.8-in diameter, what is the flow rate from each pipe?
5. A copper sphere is dropped from a building 273 ft tall. What will be its velocity on impact with the ground? Ignore air resistance.

Explore multiple problems in fluid mechanics, including calculating the average velocity in a pipe, determining the diameter of a pipe, and analyzing flow rates. Learn to apply core concepts like the continuity equation and kinematic equations for solving free-fall problems. Suitable for advanced high school and undergraduate students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation