Problem 121 - Precalculus

Question

Explain the mistake that is made. If the radius of a set of tires on a car is 15 inches and the tires rotate \(180^{\circ}\) per second, how fast is the car traveling (linear speed) in miles per hour? Solution: Write the formula for linear speed. $$\begin{aligned}&v=r \omega\\\&v=(15 \text { in. })\left(\frac{180^{\circ}}{\sec }\right) \end{aligned}$$ Let \(r=15\) inches and \(\omega=180^{\circ}\) per second. Simplify. \(v=2700 \frac{\mathrm{in}}{\mathrm{sec}}\) Let 1 mile \(=5280\) feet \(=63,360\) inches and 1 hour \(=3600\) seconds. $$v=\left(\frac{2700 \cdot 3600}{63,360}\right) \mathrm{mph}$$ Simplify. \(v \approx 153.4 \mathrm{mph}\) This is incorrect. The correct answer is approximately 2.7 mph. What mistake was made?

Step-by-Step Solution

Answer

The mistake was not using radians for angular velocity; the correct speed is approximately 2.7 mph.

Step 1: Identify the mistake in using θ

In the provided solution, the angular velocity \( \omega \) is mistakenly calculated in degrees per second instead of radians per second. In physics, angular velocity should be converted to radians for accurate calculation of linear speed.

Step 2: Convert θ to radians

Convert the given angular velocity from degrees to radians: \( 180^{\circ} = \pi \) radians. Therefore, the angular velocity \( \omega = \frac{\pi}{1} \) rad/s for 180 degrees per second.

Step 3: Recalculate linear speed using correct ω

Use the formula \( v = r \omega \) where \( r = 15 \) inches and \( \omega = \pi \) rad/s. The linear speed becomes: \( v = 15 \times \pi = 15\pi \) inches per second.

Step 4: Convert velocity to miles per hour

Convert inches per second to miles per hour: 1 mile = 63360 inches and 1 hour = 3600 seconds. Thus, \( v = 15\pi \; \text{in/sec} \times \frac{1\text{ mile}}{63360\text{ in}} \times \frac{3600\text{ sec}}{1\text{ hour}} \).

Step 5: Calculate the final value

Compute the value: \( v = 15\pi \times \frac{3600}{63360} \approx 2.7 \text{ mph} \). This correctly computes the vehicle's speed.

Key Concepts

Angular Velocity Conversion

A common mistake in calculating linear speed is not converting angular velocity from degrees to radians. Angular velocity, denoted by \( \omega \), is often given in degrees per second but needs to be in radians per second for accurate calculation. This is because radians are dimensionless, making them the preferred unit in physics for angular measurements.

To convert degrees to radians, use the formula:

• \( \text{Radians} = \text{Degrees} \times \frac{\pi}{180} \).

For example, if \( \omega = 180^{\circ} \) per second, then \( \omega = 180 \times \frac{\pi}{180} = \pi \) radians per second. This step is crucial for ensuring that subsequent calculations of linear speed are correct.

Remember, converting to radians allows you to maintain consistency with standard rotational physics calculations.

Radius Measurement

Understanding radius measurement is vital in calculating linear speed. Radius \( r \) is the distance from the center of a circle to any point on its circumference.

To apply this in real-world situations, like calculating the speed of a vehicle based on tire rotation, you need accurate radius measurements.

The radius is typically given in units like inches or centimeters, and sometimes a conversion is necessary depending on the desired final speed unit (e.g., miles per hour).

• In our example, the radius of the tire is 15 inches.

Always ensure measurements are precise because any miscalculation in radius can significantly affect the outcome of the linear speed.

Unit Conversion

After calculating the linear speed in units of length per unit time, it's often necessary to convert those units to the desired format, such as miles per hour (mph).

This involves converting both the length and time units accordingly:

• For length, you convert inches to miles knowing that 1 mile equals 63,360 inches.
• For time, convert seconds to hours as 1 hour equals 3,600 seconds.

In our exercise, we have an initial speed in inches per second, which we convert as follows: \( v = 15\pi \text{ in/sec} \times \frac{1\text{ mile}}{63,360\text{ in}} \times \frac{3600\text{ sec}}{1\text{ hour}} \).

This conversion process is essential in accurately expressing speed in a format that is practical and understandable for everyday use.

Error Analysis in Calculations

Error analysis is a crucial part of mathematical problem-solving, particularly in applications like computing speed. It involves identifying and correcting mistakes that may lead to incorrect results.

In our exercise, the error stemmed from improperly handling the units of angular velocity. This discrepancy led to an inaccurate calculation of linear speed.

Here are steps for effective error checking:

• Check the units involved at each step of your calculation.
• Ensure that all conversions are appropriately applied, especially for angular velocity from degrees to radians.
• Verify the final unit conversion to the desired measurement, such as miles per hour.

By methodically reviewing each calculation step and its unit relevance, you can pinpoint errors before they lead to incorrect conclusions.

Such diligence is important to maintain accuracy in scenarios that require precise measurements, as even small errors can lead to significant miscalculations.