If It Takes Someone $t$ Minutes To Drive, Run, Or Walk A Mile, Then Their Average Speed $s(t)$, In Miles Per Hour, Is Given By The Formula:${ S(t) = \frac{60}{t} }$ For { T \ \textgreater \ 0 $}$.Find

Introduction

Average speed is a fundamental concept in physics and mathematics, representing the rate at which an object covers a certain distance. In this article, we will delve into the mathematical formula for average speed, specifically the function $s(t) = \frac{60}{t}$, where $t$ is the time taken to cover a mile in minutes. We will explore the implications of this formula and discuss its applications in real-world scenarios.

The Formula for Average Speed

The formula for average speed is given by $s(t) = \frac{60}{t}$, where $t$ is the time taken to cover a mile in minutes. This formula can be derived from the basic definition of speed, which is the ratio of distance to time. In this case, the distance is 1 mile, and the time is $t$ minutes.

Derivation of the Formula

To derive the formula for average speed, we start with the definition of speed:

$$s = \frac{\text{distance}}{\text{time}}$$

In this case, the distance is 1 mile, and the time is $t$ minutes. Substituting these values into the formula, we get:

$$s = \frac{1 \text{ mile}}{t \text{ minutes}}$$

To convert the time from minutes to hours, we divide by 60:

$$s = \frac{1 \text{ mile}}{\frac{t}{60} \text{ hours}}$$

Simplifying the expression, we get:

$$s = \frac{60}{t}$$

Properties of the Formula

The formula for average speed has several interesting properties. First, it is a decreasing function, meaning that as the time $t$ increases, the average speed $s(t)$ decreases. This is because it takes longer to cover a mile, so the average speed is lower.

Second, the formula is undefined when $t = 0$, which makes sense because it would take an infinite amount of time to cover a mile if the time were zero.

Applications of the Formula

The formula for average speed has numerous applications in real-world scenarios. For example, it can be used to calculate the average speed of a car, a runner, or a walker. It can also be used to compare the average speeds of different modes of transportation, such as cars, buses, and trains.

Example 1: Calculating Average Speed

Suppose a person takes 10 minutes to walk a mile. What is their average speed in miles per hour?

Using the formula, we get:

$$s(t) = \frac{60}{t} = \frac{60}{10} = 6 \text{ mph}$$

Therefore, the person's average speed is 6 miles per hour.

Example 2: Comparing Average Speeds

Suppose a car takes 5 minutes to drive a mile, while a bus takes 10 minutes to drive a mile. Which mode of transportation has a higher average speed?

Using the formula, we get:

$$s_{\text{car}}(t) = \frac{60}{5} = 12 \text{ mph}$$

$$s_{\text{bus}}(t) = \frac{60}{10} = 6 \text{ mph}$$

Therefore, the car has a higher average speed than the bus.

Conclusion

In conclusion, the formula for average speed is a fundamental concept in physics and mathematics, representing the rate at which an object covers a certain distance. The formula $s(t) = \frac{60}{t}$, where $t$ is the time taken to cover a mile in minutes, has numerous applications in real-world scenarios. We have explored the implications of this formula and discussed its applications in calculating average speed and comparing average speeds of different modes of transportation.

References

• [1] "Average Speed" by Math Open Reference
• [2] "Speed" by Physics Classroom
• [3] "Distance, Speed, and Time" by Khan Academy

Further Reading

• "Motion in One Dimension" by MIT OpenCourseWare
• "Physics for Scientists and Engineers" by Paul A. Tipler
• "Mathematics for Physics and Engineering" by Frank E. Harris
  Frequently Asked Questions: Average Speed =============================================

Q: What is average speed?

A: Average speed is the rate at which an object covers a certain distance. It is calculated by dividing the distance by the time taken to cover that distance.

Q: What is the formula for average speed?

A: The formula for average speed is $s(t) = \frac{60}{t}$, where $t$ is the time taken to cover a mile in minutes.

Q: What is the unit of average speed?

A: The unit of average speed is miles per hour (mph).

Q: How is average speed different from instantaneous speed?

A: Average speed is the rate at which an object covers a certain distance over a period of time, while instantaneous speed is the rate at which an object is moving at a specific moment in time.

Q: Can average speed be greater than instantaneous speed?

A: Yes, average speed can be greater than instantaneous speed. For example, if an object is accelerating, its instantaneous speed may be lower than its average speed.

Q: Can average speed be less than instantaneous speed?

A: Yes, average speed can be less than instantaneous speed. For example, if an object is decelerating, its instantaneous speed may be higher than its average speed.

Q: How is average speed related to distance and time?

A: Average speed is directly proportional to distance and inversely proportional to time. This means that as the distance increases, the average speed increases, and as the time increases, the average speed decreases.

Q: Can average speed be zero?

A: Yes, average speed can be zero. This occurs when the object is not moving, or when the time taken to cover the distance is infinite.

Q: Can average speed be negative?

A: No, average speed cannot be negative. Average speed is always a positive value, as it represents the rate at which an object covers a certain distance.

Q: How is average speed used in real-world applications?

A: Average speed is used in a variety of real-world applications, including:

• Calculating the time it takes to travel a certain distance
• Determining the fuel efficiency of a vehicle
• Comparing the performance of different modes of transportation
• Planning routes and schedules for transportation systems

Q: What are some common mistakes to avoid when calculating average speed?

A: Some common mistakes to avoid when calculating average speed include:

• Failing to account for changes in speed over time
• Using incorrect units or values
• Ignoring the effects of acceleration or deceleration
• Failing to consider the direction of travel

Q: How can I improve my understanding of average speed?

A: To improve your understanding of average speed, try the following:

• Practice calculating average speed using different scenarios and values
• Visualize the relationship between distance, time, and average speed
• Experiment with different modes of transportation and their average speeds
• Consult with experts or resources for additional guidance and support

Conclusion

In conclusion, average speed is a fundamental concept in physics and mathematics, representing the rate at which an object covers a certain distance. By understanding the formula, properties, and applications of average speed, you can better navigate the world of physics and mathematics. Remember to avoid common mistakes and practice calculating average speed to improve your understanding of this important concept.