To find the linear speed of the Earth in feet per minute given the information that it rotates 25 revolutions per day with a distance of 9508 miles, we'll break this down step-by-step:

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a. Lawyer: Answer the following questions below

Let's address the steps to find the solution.

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b. What formula should you use to answer your problem and why not the other formula?

The formula to calculate linear speed $v$ in terms of revolutions is: $v = \frac{d \times N}{T}$

Where:

• $d$ = Distance of one revolution (the circumference)
• $N$ = Number of revolutions
• $T$ = Time (in minutes)

We don't use the formula for angular speed $\omega$, which is: $\omega = \frac{\theta}{t}$ because we are interested in linear speed, which gives us speed in terms of distance (feet per minute), rather than angle per unit time.

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c. How did you find ∅?

Since we are calculating linear speed, $\varphi$ (the angular displacement in radians) is not directly used in this problem. Instead, the key data here is the number of revolutions (25 revolutions per day). However, if we were finding angular speed, we would need to calculate $\varphi$ by converting revolutions to radians. $\varphi$ is:

\varphi = 25 \times 2\pi \quad \text{(since there are 2\pi radians in one revolution)}.

But for linear speed, $\varphi$ is not necessary.

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d. Explain your conversion method and the reasoning behind it step by step.

1. Step 1: Convert miles to feet.

   • We know that 1 mile = 5280 feet.
   • The given distance is 9508 miles.
   • Convert the distance from miles to feet: $9508 \times 5280 = 50,174,240 \, \text{feet}.$

2. Step 2: Calculate the total distance traveled in one day.

   • The Earth completes 25 revolutions in one day.
   • The total distance traveled in one day: $\text{Total distance} = 25 \times 50,174,240 = 1,254,356,000 \, \text{feet}.$

3. Step 3: Convert days to minutes.

   • One day = 24 hours, and 1 hour = 60 minutes.
   • Therefore, 1 day = $24 \times 60 = 1440 \, \text{minutes}$.

4. Step 4: Calculate the linear speed.

   • The linear speed is the total distance traveled per minute.
   • Using the formula $v = \frac{d \times N}{T}$, where $d \times N = 1,254,356,000$ feet, and $T = 1440$ minutes: $v = \frac{1,254,356,000}{1440} = 871,073.61 \, \text{feet per minute}.$

Final Answer: The linear speed of the Earth is approximately 871,073.61 feet per minute.

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Would you like further clarification on any of these steps?

Five Related Questions:

1. How would the linear speed change if the Earth completed fewer revolutions per day?
2. What is the difference between angular speed and linear speed?
3. How do you convert between angular and linear velocity?
4. How would you find the Earth's linear speed in miles per hour?
5. How does the Earth's orbital speed around the Sun compare to its rotational speed on its axis?

Tip:

Always ensure units are consistent throughout your calculations to avoid errors, especially when converting time and distance.