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Chapter 3: Problem 20

In Exercises 11-20, find the distance traveled (arc length) of a point that moves with constant speed \(v\) along a circle in time \(t\). $$ v=46 \mathrm{~km} / \mathrm{hr}, t=20 \mathrm{~min} $$

Short Answer

Expert verified
The distance traveled is 15.33 kilometers.

Step by step solution

01

Convert Time to Consistent Units

The given speed is in kilometers per hour, and time is given in minutes. We need to convert the time from minutes to hours to use the same units for calculation. Since 1 hour equals 60 minutes, we convert 20 minutes to hours: \[ t = \frac{20}{60} = \frac{1}{3} \text{ hours} \]
02

Use the Formula for Distance

The distance traveled along a circle (arc length) can be calculated using the formula \( d = v \cdot t \), where \( d \) is the distance, \( v \) is the speed, and \( t \) is the time in consistent units.
03

Substitute the Values

Substitute the values of speed \( v = 46 \text{ km/hr} \) and time \( t = \frac{1}{3} \) hour into the formula: \[ d = 46 \times \frac{1}{3} \]
04

Calculate the Arc Length

Perform the multiplication to find the arc length: \[ d = 46 \times \frac{1}{3} = \frac{46}{3} = 15.33 \text{ km} \]Thus, the distance traveled is 15.33 kilometers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Constant Speed
Understanding constant speed is crucial when it comes to calculating how far something travels over a period. **Constant speed** means that an object moves at a uniform rate, not accelerating or decelerating along its path.

This uniform rate allows us to predict movement accurately. If the speed remains unchanged, we can easily calculate the distance traveled using a straightforward multiplication of speed and time.
  • The speed, represented as \( v \), remains the same throughout the movement.
  • No matter how long or short the time, the speed does not fluctuate.
This concept is particularly useful in exercises involving the calculation of arc length or any other form of linear distance over time.
Time Conversion
When solving problems that involve motion, it's essential to ensure all units are consistent, which often means converting time into a manageable format. **Time conversion** is necessary when the same unit is not initially given, as seen in this exercise.

Here, the time given is 20 minutes, but the speed is given in kilometers per hour. To perform our calculations accurately, we convert minutes into hours.
  • Since there are 60 minutes in an hour, we can convert minutes to hours by dividing by 60.
  • For example, converting 20 minutes to hours gives us \( \frac{20}{60} = \frac{1}{3} \) hours.
This conversion aligns the time unit with the speed unit, crucial for accurate calculations.
Distance Formula
The key formula to calculate distance, especially with constant speed, is **Distance = Speed 脳 Time** (often noted as \( d = v \cdot t \)). The simplicity of this formula makes it quite powerful for determining how far an object has moved.

With everything converted into consistent units, this formula is applied directly, allowing us to track movement effectively.
  • The speed \( v \) represents the constant velocity.
  • The time \( t \) is the duration that the object travels while maintaining that speed.
In our example, we use the speed \( 46 \text{ km/hr} \) and the time \( \frac{1}{3} \) hours, substituting them into the formula to find the distance easily.
Unit Consistency
In physics and mathematics, **unit consistency** is crucial for accurate problem-solving. This means matching units across different quantities being measured.

For this exercise, ensuring the speed and time units match is imperative to properly applying the distance formula.
  • Speed is given in kilometers per hour, so time should also be in hours.
  • Any mismatch can lead to incorrect results, as the units would not align in the equation \( d = v \cdot t \).
Utilizing consistent units avoids errors, making calculations straightforward and reliable. By maintaining unit consistency, calculations like arc length or total distance embody precision and clarity.

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Most popular questions from this chapter

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