Calculating Average Speed for a Journey

To find the average speed for the whole journey, we need to determine the total distance covered and the total time taken. Since the distance is the same for both parts of the journey (going by train and returning by car), we can use a simplified formula for average speed when distances are equal.

Identify Given Speeds

• Speed of train ($v_1$): 110 km/h
• Speed of car ($v_2$): 50 km/h

Formula for Average Speed (Equal Distances)

When the distance traveled is the same for two different speeds, the average speed is calculated using the harmonic mean formula:

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

Alternatively, this can be written as:

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

Calculate Average Speed

Substitute the given speeds into the formula:

$ \text{Average Speed} = \frac{2 \times 110 \, \text{km/h} \times 50 \, \text{km/h}}{110 \, \text{km/h} + 50 \, \text{km/h}} $

Calculate the numerator:

$ 2 \times 110 \times 50 = 11000 $

Calculate the denominator:

$ 110 + 50 = 160 $

Now, divide the numerator by the denominator:

$ \text{Average Speed} = \frac{11000}{160} \, \text{km/h} $

Simplify the fraction:

$ \text{Average Speed} = \frac{1100}{16} \, \text{km/h} = \frac{275}{4} \, \text{km/h} $

Convert the fraction to a decimal:

$ \text{Average Speed} = 68.75 \, \text{km/h} $

Conclusion

The average speed for the whole journey is 68.75 km/h.