Motor Car Average Speed Calculation

This problem involves calculating the average speed when the journey is divided into two equal distances covered at different speeds.

Understanding Average Speed

Average speed is defined as the total distance traveled divided by the total time taken.

Formula: Average Speed = Total Distance / Total Time

Applying the Formula

Let the total distance be D. The first half distance is $D/2$ and the second half distance is $D/2$. Let the speeds be $v_1 = 30$ km/h for the first half and $v_2 = 50$ km/h for the second half.

Time taken for the first half ($t_1$) = Distance / Speed = $\frac{D/2}{v_1} = \frac{D}{2 \times 30}$

Time taken for the second half ($t_2$) = Distance / Speed = $\frac{D/2}{v_2} = \frac{D}{2 \times 50}$

Total Time ($T$) = $t_1 + t_2 = \frac{D}{60} + \frac{D}{100}$

To add these fractions, find a common denominator (300):

$T = \frac{5D}{300} + \frac{3D}{300} = \frac{8D}{300}$

Now, calculate the Average Speed:

Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}} = \frac{D}{\frac{8D}{300}}$

Average Speed = $D \times \frac{300}{8D} = \frac{300}{8}$

Average Speed = $37.5$ km/h

Alternative Method (Harmonic Mean)

For cases where two equal distances are covered at different speeds ($v_1$ and $v_2$), the average speed is the harmonic mean:

Average Speed $= \frac{2}{\frac{1}{v_1} + \frac{1}{v_2}}$

Substitute the given speeds:

Average Speed $= \frac{2}{\frac{1}{30} + \frac{1}{50}}$

Average Speed $= \frac{2}{\frac{5 + 3}{150}} = \frac{2}{\frac{8}{150}}$

Average Speed $= 2 \times \frac{150}{8} = \frac{300}{8} = 37.5$ km/h

Conclusion

The average speed of the motor car is 37.5 km/h.