Let the speed of the faster train be $x$ m/s and the speed of the slower train be $y$ m/s.

Step 1: Form the First Equation

The total distance to be covered for overtaking is the sum of the lengths of the two trains:
$130 + 110 = 240 \text{ m}$
Since they are moving in the same direction, their relative speed is:
$(x - y) \text{ m/s}$
The time taken to overtake is 15 seconds, so we set up the equation:
$\frac{240}{x - y} = 15$
$x - y = \frac{240}{15} = 16$

Step 2: Form the Second Equation

When the slower train moves at half its speed, its new speed is $\frac{y}{2}$.
The new relative speed is:
$x - \frac{y}{2}$
The time taken for overtaking in this case is 10 seconds:
$\frac{240}{x - \frac{y}{2}} = 10$
$x - \frac{y}{2} = \frac{240}{10} = 24$

Step 3: Solve for $x$ and $y$

We now have the system of equations:

1. $x - y = 16$
2. $x - \frac{y}{2} = 24$

Solve for $y$:

Rewriting the second equation:
$x = 24 + \frac{y}{2}$
Substituting $x = y + 16$ into this equation:
$y + 16 = 24 + \frac{y}{2}$
Multiply everything by 2 to eliminate the fraction:
$2y + 32 = 48 + y$
$2y - y = 48 - 32$
$y = 16$

Solve for $x$:

$x = 16 + 16 = 32$

Final Answer:

• Speed of the faster train: 32 m/s
• Speed of the slower train: 16 m/s

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Here are 5 related questions to explore further:

1. What would be the time taken for the faster train to overtake the slower train if they were moving in opposite directions?
2. How would the overtaking time change if the lengths of the trains were different?
3. What would be the speed of the slower train if the overtaking time was 20 seconds instead of 15?
4. How does the concept of relative speed apply to other real-life scenarios like river currents and airplane wind resistance?
5. If the trains were accelerating instead of moving at constant speeds, how would the calculation change?

Tip: Always remember that in problems involving relative motion in the same direction, subtract the speeds; in opposite directions, add them. 🚆