Question

A vehicle travels half the distance with speed θ and the remaining distance with speed 2θ. its average speed is

• A. \(\frac{3\theta}{3}\)
• B. \(\frac{\theta}{3}\)
• C. \(\frac{2\theta}{3}\)
• D. \(\frac{4\theta}{3}\)

Answer 1

The Correct Option is D

To find the average speed of the vehicle when it travels half the distance with speed \(\theta\) and the other half with speed \(2\theta\), we can use the concept of average speed based on time taken for each segment of the journey.

Step-by-step Solution:

1. Consider the total distance traveled is \(D\). Since the vehicle travels half the distance with each speed, each segment is \(D/2\).
2. Calculate the time taken for the first half:

The time taken \(t_1\) to travel the first half \(D/2\) with speed \(\theta\) is:

\(t_1 = \frac{D/2}{\theta} = \frac{D}{2\theta}\)

1. Calculate the time taken for the second half:

The time taken \(t_2\) to travel the second half \(D/2\) with speed \(2\theta\) is:

\(t_2 = \frac{D/2}{2\theta} = \frac{D}{4\theta}\)

1. Total time taken for the journey:

\(t_{\text{total}} = t_1 + t_2 = \frac{D}{2\theta} + \frac{D}{4\theta}\)

To simplify, find a common denominator:

\(t_{\text{total}} = \frac{2D}{4\theta} + \frac{D}{4\theta} = \frac{3D}{4\theta}\)

1. Calculate the average speed:

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

\(\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{3D}{4\theta}}\)

Canceling \(D\) from the numerator and the denominator gives:

\(= \frac{4\theta}{3}\)

Thus, the average speed of the vehicle is \(\frac{4\theta}{3}\).