Question 34: A vehicle travels half the distance with speed \vartheta and the remaining distance with speed 2 \vartheta. Its average speed is:
(1) \frac{4 \vartheta}{3}
(2) \frac{3 \vartheta}{4}
(3) \frac{\vartheta}{3}
(4) \frac{2 \vartheta}{3}
Answer: Option (1)
Explanation:
Let the total distance travelled be 2d.
The vehicle travels distance d with speed \vartheta
and distance d with speed 2\vartheta.
Time taken to cover first half is t_{1}=\frac{d}{\vartheta}.
Time taken to cover second half is t_{2}=\frac{d}{2\vartheta}.
Total time taken is t=t_{1}+t_{2}=\frac{d}{\vartheta}+\frac{d}{2\vartheta}=\frac{3d}{2\vartheta}.
Average speed is defined as total distance divided by total time.
Thus, average speed is v_{\text{avg}}=\frac{2d}{\frac{3d}{2\vartheta}}.
On simplification, v_{\text{avg}}=\frac{4\vartheta}{3}.
Hence, the correct answer is Option (1).