Question

A car travels a distance of ' $x$ ' with speed $v_1$ and then same distance ' $x$ ' with speed $v_2$ in the same direction The average speed of the car is:

• A. $\frac{v_1 v_2}{2\left(v_1+v_2\right)}$
• B. $\frac{2 x}{v_1+v_2}$
• C. $\frac{2 v_1 v_2}{v_1+v_2}$
• D. $\frac{v_1+v_2}{2}$

Answer 1

The Correct Option is C

To determine the average speed of a car traveling the same distance ' \(x\) ' with two different speeds, we must first understand the formula for average speed when distances are equal. The average speed for such a journey is calculated by considering the total distance traveled and the total time taken.

1. First, compute the time taken to travel distance ' \(x\) ' with speed \(v_1\). The time, \(t_1\), is given by: \(t_1 = \frac{x}{v_1}\)
2. Next, compute the time taken to travel the same distance ' \(x\) ' with speed \(v_2\). The time, \(t_2\), is given by: \(t_2 = \frac{x}{v_2}\)
3. The total distance traveled is \(2x\).
4. The total time taken for the journey is the sum of \(t_1\) and \(t_2\): \(t_{\text{total}} = \frac{x}{v_1} + \frac{x}{v_2} = x\left(\frac{1}{v_1} + \frac{1}{v_2}\right)\)
5. Thus, the average speed \(v_{\text{avg}}\) is calculated as: \(v_{\text{avg}} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{2x}{x\left(\frac{1}{v_1} + \frac{1}{v_2}\right)}\)
6. Simplifying this expression, we get: \(v_{\text{avg}} = \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}} = \frac{2v_1v_2}{v_1+v_2}\)

Therefore, the average speed of the car when it travels two equal distances with different speeds is \(\frac{2 v_1 v_2}{v_1+v_2}\). This matches the correct answer.