Understanding Average Speed for a Round Trip

The question asks about the average speed of a car during a whole journey, which involves traveling from point A to point B and then returning from B to A. The car has different speeds for each leg of the journey: 40 Km/h for A to B and 60 Km/h for B to A.

It is important to remember that average speed is defined as the total distance traveled divided by the total time taken. When the distances traveled in two different speeds are equal (like in a round trip between two points), the simple average (arithmetic mean) of the speeds is not the correct way to calculate the average speed.

Calculating Average Speed for Equal Distances

Let the distance between points A and B be \(d\). The time taken to travel from A to B with speed \(v_1 = 40\) Km/h is \(t_1 = \frac{d}{v_1} = \frac{d}{40}\) hours.

The time taken to return from B to A with speed \(v_2 = 60\) Km/h is \(t_2 = \frac{d}{v_2} = \frac{d}{60}\) hours.

The total distance traveled during the whole journey (A to B and B to A) is \(D = d + d = 2d\).

The total time taken for the whole journey is \(T = t_1 + t_2 = \frac{d}{40} + \frac{d}{60}\).

To add the fractions for total time, find a common denominator for 40 and 60, which is 120.

\(T = \frac{d \times 3}{40 \times 3} + \frac{d \times 2}{60 \times 2} = \frac{3d}{120} + \frac{2d}{120} = \frac{3d + 2d}{120} = \frac{5d}{120} = \frac{d}{24}\) hours.

Now, the average speed is the total distance divided by the total time:

\(\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{d}{24}}\)

To divide by a fraction, multiply by its reciprocal:

\(\text{Average Speed} = 2d \times \frac{24}{d}\)

The distance \(d\) cancels out:

\(\text{Average Speed} = 2 \times 24 = 48\) Km/h.

Using the Formula for Average Speed (Equal Distances)

Alternatively, for a journey covering two equal distances with speeds \(v_1\) and \(v_2\), the average speed can be calculated directly using the formula:

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

Given \(v_1 = 40\) Km/h and \(v_2 = 60\) Km/h.

\(\text{Average Speed} = \frac{2 \times 40 \times 60}{40 + 60}\)

\(\text{Average Speed} = \frac{2 \times 2400}{100}\)

\(\text{Average Speed} = \frac{4800}{100}\)

\(\text{Average Speed} = 48\) Km/h.

Both methods yield the same result. The average speed of the car during the whole journey is 48 Km/h.

Comparison of Average Speeds

Let's look at the different speeds involved:

• Speed from A to B: 40 Km/h
• Speed from B to A: 60 Km/h
• Simple average (arithmetic mean): \(\frac{40 + 60}{2} = 50\) Km/h
• Calculated average speed: 48 Km/h

The calculated average speed (48 Km/h) is less than the simple average (50 Km/h). This is because the car spends more time traveling at the slower speed (40 Km/h) over the equal distance compared to the time spent traveling at the faster speed (60 Km/h).

Revision Table: Average Speed Concepts

Additional Information: Speed vs. Velocity

While speed and velocity are often used interchangeably in everyday language, they are distinct concepts in physics:

• Speed: Speed is a scalar quantity that measures how fast an object is moving. It is the magnitude of the rate of motion. Speed only considers the distance covered.
• Velocity: Velocity is a vector quantity that describes both the speed and the direction of an object's motion. It is the rate of change of displacement.

In this problem, the car travels from A to B and returns to A. The total distance traveled is twice the distance between A and B. However, the total displacement is zero because the car ends up at its starting point. Therefore, while the car has a non-zero average speed, its average velocity for the entire round trip would be zero.

The question specifically asks for average speed, which is what we calculated.