Understanding Relative Speed

When objects move in the same direction, their relative speed is the difference between their speeds. In this case, the relative speed between the train and each person is the speed of the train minus the speed of the person. The distance the train covers to overtake a person is equal to the length of the train.

Converting Speeds

First, convert the speeds of the persons from km/hr to m/s, as the time is given in seconds.

• Person 1 speed: $9 \text{ km/hr} = 9 \times \frac{5}{18} \text{ m/s} = 2.5 \text{ m/s}$
• Person 2 speed: $12.6 \text{ km/hr} = 12.6 \times \frac{5}{18} \text{ m/s} = 3.5 \text{ m/s}$

Setting Up Equations

Let the speed of the train be $S_t$ m/s and the length of the train be $L$ meters.

The relative speed when overtaking the first person is $(S_t - 2.5)$ m/s. The time taken is 13.5 seconds.

The relative speed when overtaking the second person is $(S_t - 3.5)$ m/s. The time taken is 15 seconds.

Using the formula: Distance = Speed × Time

• For Person 1: $L = (S_t - 2.5) \times 13.5$ (Equation 1)
• For Person 2: $L = (S_t - 3.5) \times 15$ (Equation 2)

Calculating Train Speed

Since the length of the train ($L$) is the same in both cases, we can equate Equation 1 and Equation 2:

$(S_t - 2.5) \times 13.5 = (S_t - 3.5) \times 15$

Expand the equation:

$13.5 S_t - (2.5 \times 13.5) = 15 S_t - (3.5 \times 15)$

$13.5 S_t - 33.75 = 15 S_t - 52.5$

Rearrange the terms to solve for $S_t$:

$52.5 - 33.75 = 15 S_t - 13.5 S_t$

$18.75 = 1.5 S_t$

$S_t = \frac{18.75}{1.5} = 12.5 \text{ m/s}$

Converting Back to km/hr

Convert the train's speed from m/s back to km/hr:

$S_t = 12.5 \text{ m/s} = 12.5 \times \frac{18}{5} \text{ km/hr} = 45 \text{ km/hr}$