Question

A covers hall of his distance with \(20 \mathrm{~km} / \mathrm{h}\) and rest with \(30 \mathrm{~km} / \mathrm{h}\). What is the average speed during the whole journey? (a) \(20 \mathrm{~km} / \mathrm{h}\) (c) \(25 \mathrm{~km} / \mathrm{h}\) (d) \(26 \mathrm{~km} / \mathrm{h}\)

Short answer

Answer: The average speed during the whole journey is 24 km/h.

Step-by-step solution

Assign variables and find time taken to cover each part of journey

Let the total distance be \(d\). Half of the distance will then be \(d/2\). To find the time taken to cover each part of the journey, we'll use the formula: time = distance/speed. Time taken to cover the first part of the journey (at 20 km/h): \(t_1 = \frac{d}{2} \div \mathrm{20~km/h} = \frac{d}{40~h}\) Time taken to cover the second part of the journey (at 30 km/h): \(t_2 = \frac{d}{2} \div \mathrm{30~km/h} = \frac{d}{60~h}\)

Find the total time taken for the journey

Add the times for both parts of the journey: Total time \(t = t_1 + t_2 = \frac{d}{40~h} + \frac{d}{60~h} = \frac{3d+2d}{120~h} = \frac{5d}{120~h}\)

Calculate the average speed

Average speed is the total distance divided by the total time. We'll use the formula: average speed = total distance/total time. Average speed = \(\frac{d}{\frac{5d}{120~h}}\) By simplifying the above expression, we get: Average speed = \(\frac{120}{5} \mathrm{~km/h} = 24 \mathrm{~km/h}\) So, the average speed during the whole journey is \(\boxed{24 \mathrm{~km/h}}\). Note that this option is not among the alternatives provided, which indicates that there might be a mistake in the given options.

Key concepts

Speed and Distance

Understanding the relationship between speed and distance is crucial in solving motion problems efficiently. Speed is defined as how fast an object is traveling per unit of time. It is computed using the formula:

• Speed = Distance / Time

Distance is the length of the path traveled by an object. In motion problems, it is often divided into sections to simplify calculations, especially if the speed changes midway through the journey.
In the given problem, the journey is split into two equal parts, each covered at different speeds. The first half is at a speed of 20 km/h, and the second at 30 km/h.
This approach highlights a practical understanding, as it provides clarity on how to break down a problem where variables change over the course of an event.
By dividing the journey into different speed segments, we can simplify the calculation process.

Time Calculation

Time calculation is pivotal in determining the efficiency of a trip or journey. It represents the measure of how long an object or person takes to cover a certain distance at a designated speed. The formula to calculate time is:

• Time = Distance / Speed

In this exercise, we split the total distance, denoted by "d," into two equal halves. Each half requires a distinct calculation since it is traveled at a different speed. Calculating these separately helps in precisely assessing the total time taken for the journey.
For the first half, at 20 km/h, the time calculation is performed as:

• Time taken to cover the first half = \( \frac{d}{2} \div 20 \) = \( \frac{d}{40} \) hours

For the second half, at 30 km/h, it is:

• Time taken to cover the second half = \( \frac{d}{2} \div 30 \) = \( \frac{d}{60} \) hours

Summing these two time values gives us the total time which is crucial for calculating average speed. Thoroughly understanding this section is essential for effectively solving motion problems.

Motion Problems

Motion problems often involve calculating different attributes like speed, distance, or time, usually when one of them varies during the journey. These problems test the capacity to integrate multiple concepts and formulas to arrive at a final answer.
The key in motion problems is to correctly align these variables according to the changes occurring, such as different speeds as depicted in this exercise.
Average speed is particularly a central concept in motion problems. It is different from simply taking an arithmetic mean of the speeds. It is calculated by:

• Average Speed = Total Distance / Total Time

In the provided exercise, solving for average speed requires not only understanding each leg of the journey with individual speeds but also being adept at determining and combining the total time for the whole journey under changing conditions.
This example helps clarify that the average speed does not equal the arithmetic average of 20 km/h and 30 km/h, emphasizing the importance of proper calculation techniques in achieving the right answer.