Calculating Average Speed for Journey and Return

This question asks us to find the average speed of a man who travels a certain distance at one speed and returns the same distance at a different speed. The speeds are given as algebraic expressions involving \(x\).

The key to solving this is understanding the formula for average speed, especially in the context of a round trip where the distance for the outbound and return journeys is the same.

Formula for Average Speed on a Round Trip

When an object travels a distance \(d\) at speed \(v_1\) and returns the same distance \(d\) at speed \(v_2\), the total distance traveled is \(d + d = 2d\). The time taken for the first part is \(t_1 = \frac{d}{v_1}\) and for the return part is \(t_2 = \frac{d}{v_2}\). The total time taken is \(t_1 + t_2 = \frac{d}{v_1} + \frac{d}{v_2}\).

Average speed is defined as the total distance divided by the total time.

$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} $$

Substituting the values:

$$ \text{Average Speed} = \frac{2d}{\frac{d}{v_1} + \frac{d}{v_2}} $$

We can factor out \(d\) from the denominator:

$$ \text{Average Speed} = \frac{2d}{d \left(\frac{1}{v_1} + \frac{1}{v_2}\right)} $$

Canceling \(d\) from the numerator and denominator (assuming \(d \neq 0\)):

$$ \text{Average Speed} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} $$

This formula is very useful for round trips or any situation where equal distances are covered at different speeds. It can also be written as:

$$ \text{Average Speed} = \frac{2v_1 v_2}{v_1 + v_2} $$

We will use the form \( \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \) as it might simplify calculations involving fractions directly.

Applying the Speeds to the Average Speed Formula

Given speeds are:

• Speed for the journey (\(v_1\)): \( \frac{1}{x} \) km/h
• Speed for the return (\(v_2\)): \( \frac{1}{x^2} \) km/h

Now, substitute these speeds into the average speed formula:

$$ \text{Average Speed} = \frac{2}{\frac{1}{\left(\frac{1}{x}\right)} + \frac{1}{\left(\frac{1}{x^2}\right)}} $$

Simplifying the denominators within the main denominator:

$$ \frac{1}{\left(\frac{1}{x}\right)} = 1 \times \frac{x}{1} = x $$

$$ \frac{1}{\left(\frac{1}{x^2}\right)} = 1 \times \frac{x^2}{1} = x^2 $$

Substitute these back into the average speed formula:

$$ \text{Average Speed} = \frac{2}{x + x^2} $$

Comparing Calculated Average Speed with Options

The calculated average speed is \( \frac{2}{x + x^2} \) km/h.

Let's look at the given options:

• Option 1: \( \frac{x+1}{x^2} \)
• Option 2: \( \frac{2}{x+x^2} \)
• Option 3: \( \frac{1}{x+x^2} \)
• Option 4: \( \frac{2x^2}{x+1} \)

Our calculated average speed \( \frac{2}{x+x^2} \) matches Option 2.

Step-by-Step Calculation Summary

Here is a summary of the steps:

1. Identify the speeds \(v_1 = \frac{1}{x}\) and \(v_2 = \frac{1}{x^2}\).
2. Use the average speed formula for equal distances: \( \text{Average Speed} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \).
3. Substitute the given speeds: \( \text{Average Speed} = \frac{2}{\frac{1}{\left(\frac{1}{x}\right)} + \frac{1}{\left(\frac{1}{x^2}\right)}} \).
4. Simplify the reciprocal terms in the denominator: \( \frac{1}{\left(\frac{1}{x}\right)} = x \) and \( \frac{1}{\left(\frac{1}{x^2}\right)} = x^2 \).
5. Substitute the simplified terms back: \( \text{Average Speed} = \frac{2}{x + x^2} \).
6. Compare the result with the options.

Revision Table: Speed, Distance, and Time Concepts

Additional Information: Understanding Speed and Time

Speed, distance, and time are fundamental concepts in physics and mathematics. They are interconnected. If you know any two, you can find the third using the formulas. Average speed is not always simply the average of the speeds (arithmetic mean), especially when different speeds are maintained for different durations or over different distances.

In this specific problem, because the distance for the journey and the return is the same, the time taken for each leg is different (unless \(x = x^2\), which implies \(x=1\)). Since the time taken for each part is different, the simple arithmetic mean of the speeds (\(\frac{v_1 + v_2}{2}\)) would not give the correct average speed over the entire journey. The formula we used, \( \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} \), which is also known as the harmonic mean of the speeds, is appropriate when equal distances are covered at different speeds.

Understanding when to use the harmonic mean versus the arithmetic mean for calculating average speed is crucial. Use the harmonic mean when distances are equal but speeds/times vary. Use the arithmetic mean only if the time intervals are equal for each speed.