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

The average speed of a vehicle is calculated as the total distance traveled divided by the total time taken. Since the vehicle travels half the distance with speed \( \theta \) and the remaining half with speed \( 2\theta \), let's denote the total distance as \( 2d \). This means it travels \( d \) at each speed.

1. Calculate Time Taken for Each Half:
- Time for first half: \( t_1 = \frac{d}{\theta} \)
- Time for second half: \( t_2 = \frac{d}{2\theta} \)

2. Total Time Taken:
\( t_{\text{total}} = t_1 + t_2 = \frac{d}{\theta} + \frac{d}{2\theta} \)

3. Simplify Total Time: 
\( t_{\text{total}} = \frac{2d + d}{2\theta} = \frac{3d}{2\theta} \)

4. Average Speed Formula:
\( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2d}{\frac{3d}{2\theta}} \)

5. Simplify Average Speed:
\(\text{Average Speed} = \frac{2d \times 2\theta}{3d} = \frac{4\theta}{3}\)

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