Calculating Average Speed for Half Distance with Different Speeds: A Comprehensive Guide

Introduction

Calculating the average speed for a journey involving different speeds for different halves of the distance is a common problem in mathematics. This article will provide a step-by-step guide to solving such problems, detailing the calculations and the reasoning behind the methods used.

Understanding the Problem

A motorist travels half of a journey at 40 km/hr and the other half at 60 km/hr. We need to find the average speed for the entire journey.

Case 1: Using Time as the Measure

Let the total distance be (d) km. The first half of the journey is covered at 40 km/hr, and the second half at 60 km/hr.

Step 1: Calculate the time taken for each half.

Time for first half: (t_1 frac{d/2}{40} frac{d}{80}) hours

Time for second half: (t_2 frac{d/2}{60} frac{d}{120}) hours

Total time: (T t_1 t_2 frac{d}{80} frac{d}{120} frac{3d 2d}{240} frac{5d}{240} frac{d}{48}) hours

Step 2: Calculate the average speed.

Using the formula ( text{Average speed} frac{text{Total distance}}{text{Total time}} )

( text{Average speed} frac{d}{frac{d}{48}} 48 ) km/hr

Case 2: Using Distance as the Measure

Alternatively, let's consider the journey is divided into two equal distances rather than two equal time intervals.

Step 1: Calculate the time taken for each distance.

Time for first half at 60 km/hr: (t_1 frac{d}{2 times 60} frac{d}{120}) hours

Time for second half at 40 km/hr: (t_2 frac{d}{2 times 40} frac{d}{80}) hours

Total time: (T t_1 t_2 frac{d}{120} frac{d}{80} frac{2d 3d}{240} frac{5d}{240} frac{d}{48}) hours

Step 2: Calculate the average speed.

Using the formula ( text{Average speed} frac{text{Total distance}}{text{Total time}} )

( text{Average speed} frac{d}{frac{d}{48}} 48 ) km/hr

Note that in this case, we are dividing the distance into two equal parts.

Two Different Answers: Half the Time or Half the Distance

The problem can be interpreted in two different ways: either half the time or half the distance. Let's explore each interpretation:

Half the time: If the journey is divided into two equal time intervals, the average speed can be calculated by taking the arithmetic mean of the two speeds.

(text{Average speed} frac{60 40}{2} frac{100}{2} 50 ) km/hr

Half the distance: If the journey is divided into two equal distances, the average speed is calculated using the harmonic mean of the two speeds.

(text{Average speed} frac{2}{frac{1}{60} frac{1}{40}} frac{2}{frac{1}{60} frac{1}{40}} frac{2}{frac{1}{60} frac{3}{120}} frac{2}{frac{2 3}{120}} frac{2}{frac{5}{120}} frac{2 times 120}{5} 48 ) km/hr

The average speed is 48 km/hr if the journey is divided into two equal distances.

Real-World Application: Rideshare Services

Consider a rideshare service, such as Uber or Lyft, that charges both by the minute and the mile. The problem of determining the average speed can be updated to calculate fares based on half the distance at 60 mph and the other half at 40 mph. This would need additional information about the individual per-minute and per-mile rates but would be a more challenging problem to solve.

Conclusion

The average speed for a journey involving different speeds for different halves of the distance can be calculated using the harmonic mean when the distance is split equally or the arithmetic mean when the time is split equally. The choice of method depends on how the journey is divided.

Keywords: average speed, half distance, different speeds, harmonic mean, arithmetic mean