Question

A car covers one third part of its straight path with speed \(\mathrm{V}_{1}\) and the rest with speed \(\mathrm{V}_{2}\). What is its average speed? (A) $\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+\mathrm{v}_{2}\right)\right]$ (B) $\left[\left(2 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(3 \mathrm{v}_{1}+\mathrm{v}_{2}\right)\right]$ (C) $\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(\mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]$ (D) $\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]$

Short answer

The short answer is: \(\boxed{\text{(D)}\:\frac{3V_1V_2}{2V_1+2V_2}}\).

Step-by-step solution

Calculate the time taken for each part

Let \(d_1\) be the distance covered with speed \(V_1\), and \(d_2\) be the distance covered with speed \(V_2\). We know that \(d_1 = \frac{1}{3}d\) (where \(d\) is the total distance), and \(d_2 = \frac{2}{3}d\). Now, using the formula of time, \[Time = \frac{Distance}{Speed}\] Calculate the time taken for each part: \[t_1 = \frac{d_1}{V_1} = \frac{\frac{1}{3}d}{V_1}\] \[t_2 = \frac{d_2}{V_2} = \frac{\frac{2}{3}d}{V_2}\]

Calculate Total Time

Add the time taken for each part to calculate the total time: \[Total\:Time = t_1 + t_2 = \frac{\frac{1}{3}d}{V_1} + \frac{\frac{2}{3}d}{V_2}\]

Calculate the Average Speed

Now, using the formula for the average speed and the total time calculated in step 2, \[Average\:Speed = \frac{Total\:Distance}{Total\:Time} = \frac{d}{\frac{\frac{1}{3}d}{V_1} + \frac{\frac{2}{3}d}{V_2}}\]

Simplify the formula and compare to the given options

\[Average\:Speed = \frac{d}{\frac{\frac{1}{3}d}{V_1} + \frac{\frac{2}{3}d}{V_2}} \cdot \frac{3V_1V_2}{3V_1V_2}\] \[Average\:Speed = \frac{3V_1V_2d}{\left(\frac{1}{3}dV_2\right) + \left(\frac{2}{3}dV_1\right)}\] \[Average\:Speed = \frac{3V_1V_2d}{\left(V_2d\right) + \left(2V_1d\right)}\] Since the total distance (d) is the same for both parts, we can cancel it out: \[Average\:Speed = \frac{3V_1V_2}{\left(V_2\right) + \left(2V_1\right)}\] Comparing to the given options, the final answer is (D) \(\left[\left(3 \mathrm{v}_{1} \mathrm{v}_{2}\right) /\left(2 \mathrm{v}_{1}+2 \mathrm{v}_{2}\right)\right]\).