Question

In a well-known novel a person travels around the world in 80 days. (a) What is the person's approximate average speed during the adventure? (b) What is the approximate average velocity for the entire trip? (Note that Earth's circumference at the equator is \(40,075 \mathrm{~km}\).)

Short answer

The average speed is 20.88 km/h, and the average velocity is 0 km/h.

Step-by-step solution

Understand the Problem

To find the average speed and average velocity, we need to use the total distance and time. Average speed is defined as total distance divided by total time, while average velocity is total displacement divided by total time.

Identify Key Values

The total distance traveled is the circumference of Earth, which is given as 40,075 km. The total time taken for the trip is 80 days.

Convert Time to Hours

Since speed is often given in km/h, convert the time from days to hours. There are 24 hours in a day, so:\[80 ext{ days} \times 24 ext{ hours/day} = 1920 ext{ hours}\]

Calculate Average Speed

Average speed is calculated as total distance divided by total time. Use the formula:\[\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}\]Substitute the known values:\[\text{Average speed} = \frac{40,075 ext{ km}}{1920 ext{ hours}} \approx 20.88 ext{ km/h}\]

Calculate Average Velocity

For a complete round trip around the world, the displacement is zero, since the start and end points are the same. Average velocity is calculated as the displacement divided by total time. Thus:\[\text{Average velocity} = \frac{0 ext{ km}}{1920 ext{ hours}} = 0 ext{ km/h}\]

Conclude the Exercise

The approximate average speed during the trip is 20.88 km/h. The average velocity, due to zero net displacement, is 0 km/h.

Key concepts

Distance and Displacement

When we talk about distance, we're referring to the total path traveled, regardless of direction. In the context of traveling around the Earth, the distance would be the entire circumference of the Earth. This means traveling from the starting point, all the way around the Earth, and back to the starting point. In this case, the distance is given as 40,075 kilometers.

On the other hand, displacement is a bit different. Displacement is concerned with the overall change in position, measured in a straight line from start to finish. So, if you start and end at the same point, like in a round-the-world trip, your displacement is zero. Even though the traveler covered a huge distance, they didn’t change their final position compared to the start. This is an essential distinction when calculating average speed and velocity.

• Distance: total path traveled; here, 40,075 km.
• Displacement: overall change in position; here, 0 km.

Please remember, while distance involves only the magnitude, displacement also takes into account the direction.

Circumference of Earth

The circumference of the Earth is the total distance around the equator. This is a crucial piece of information when calculating journeys like traveling around the world. In this particular problem, the circumference of the Earth is provided as 40,075 kilometers. This value is used to compute the average speed of the traveler who goes around the Earth.

Knowing this circumference allows us to calculate the total distance covered during a circumnavigation. It's especially helpful in real-world calculations, such as travel times and distances for flights and global shipping routes. The given task utilizes this fixed value to find the average speed over a set time frame.

• The Earth's circumference is measured as approximately 40,075 kilometers at the equator.
• This measurement is foundational in calculating global travel distances, like in the example of the around-the-world trip.

The constancy of this value makes it a helpful standard for travelers and scientists alike.

Conversion of Units

In physics and everyday calculations, being proficient in converting units is critical. Here, our task entails converting days to hours to ensure consistent units across speed calculations. In the problem, the traveler circumnavigates the Earth in 80 days. Since speed is often expressed in terms of kilometers per hour (km/h), we must convert our time unit from days to hours.

To perform this conversion, we multiply the number of days by the number of hours in a day. For this exercise: \[80 \text{ days} \times 24 \text{ hours/day} = 1920 \text{ hours} \]This conversion allows us to calculate average speed accurately as it maintains uniformity between distance and time units.

• Multiply the number of days by 24 to convert to hours.
• Conversion ensures that time units align correctly with distance measurements for speed calculations.

Such consistent unit conversions are vital to avoid errors in calculations and to maintain clarity in results.