Q. A bus covers the first half of a certain distance with speed v₁ and the second half with a speed v₂. What is the average speed during the whole journey?

Correct Answer: b) 2v1v2 / (v1 + v2)

Explanation:

The average speed is defined as the total distance traveled divided by the total time taken.

1. Let the total distance be “D”.

– The first half of the distance is D/2, and the second half is also D/2.

2. Time taken to cover the first half of the distance:

Speed = Distance / Time, so Time = Distance / Speed.

For the first half, the time taken is:

t1 = D/2 / v1 = D / 2v1

3. Time taken to cover the second half of the distance:

Similarly, for the second half, the time taken is:

t2 = D/2 / v2 = D / 2v2

4. Total time for the journey:

The total time is the sum of t1 and t2:

t_total = t1 + t2 = D / 2v1 + D / 2v2

Taking a common denominator:

t_total = D(v2 + v1) / 2v1v2

5. Average speed formula:

The average speed is given by:

Average speed = Total distance / Total time

Substituting the values:

Average speed = D / t_total = D / (D(v1 + v2) / 2v1v2)

Simplify the expression:

Average speed = 2v1v2 / (v1 + v2)

Final Answer: The average speed during the whole journey is: b) 2v1v2 / (v1 + v2)