Calculating Average Velocity When Differing Velocities Are Maintained for Equal Distances

Calculating the average velocity of a car that travels equal distances at different speeds is a common but nuanced problem in physics and can be particularly challenging when the time spent at each speed is not uniform. In this article, we will explore how to calculate the average velocity in such scenarios and discuss the importance of considering time durations and not just the arithmetic mean of velocities.

Example and Problem Statement

A car travels equal distances of 60 km with velocities of 60 km/h, 20 km/h, and 10 km/h, respectively. We want to find the magnitude of the average velocity of the car over the entire journey. Let's break it down step by step.

Step-by-Step Solution

First, we calculate the time taken for each segment of the journey.

For the first 60 km at 60 km/h, the time taken is 1 hour. For the second 60 km at 20 km/h, the time taken is 3 hours. For the third 60 km at 10 km/h, the time taken is 6 hours.

Now, let's sum up the distances and times.

Total distance 60 km 60 km 60 km 180 km Total time 1 hour 3 hours 6 hours 10 hours

The average velocity is calculated as the total distance divided by the total time.

Average velocity Total distance / Total time 180 km / 10 hours 18 km/h

A Common Misconception and Correct Approach

It's tempting to simply average the velocities (60 20 10) / 3 30 km/h. However, this would be incorrect due to the different times spent at each speed. Average velocity is given by the total distance divided by the total time, which accounts for the varying durations.

Using a Common Constant Distance for Simplification

For instructional purposes, we can use a common distance, say 60 km, and express the time in terms of a constant K.

Let's assume the car travels 60 km each at the respective speeds. Here, the time for each segment can be calculated as follows:

First segment: 60 km / 60 km/h 1 hour Second segment: 60 km / 20 km/h 3 hours Third segment: 60 km / 10 km/h 6 hours

Thus, the total time 10 hours.

The total distance for three segments is 180 km. Therefore, the average velocity is 180 km / 10 hours 18 km/h.

General Approach for Varying Distances

Suppose the total distance is 3D in three segments. The time for each segment can be expressed as D/speed. So, the total time is D/60 D/20 D/10 D(1/60 3/60 6/60) D/6 hours. The average velocity is 3D / (D/6) 18 km/h.

Important Point to Note

The average velocity depends on the time spent at each speed, not just the average of the velocities. If the car spends more time at the slower speeds, the average velocity will be lower than the arithmetic mean of the velocities.

Conclusion

In conclusion, calculating the average velocity of a car that travels equal distances at varying speeds is a straightforward problem once the time components are taken into account. The correct approach involves summing the distances and times appropriately and then dividing the total distance by the total time.

Understanding this concept helps in accurately analyzing motion scenarios and makes it easier to apply this knowledge in more complex real-world situations. Whether you're a physics student or an engineer, this method of calculation is invaluable for solving problems involving velocity and motion.

Keyword1: average velocity Keyword2: equal distances Keyword3: varying velocities