Places A and B are 100 Km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, theymeet in Shours. If they travel towards each other, they meet in hour.

Let's define the problem with the given information:

1. The distance between places A and B is 100 km.
2. When the cars travel in the same direction, they meet after \( S \) hours.
3. When the cars travel towards each other, they meet after \( T \) hours.

Let:
- \( v_1 \) be the speed of the car starting from A.
- \( v_2 \) be the speed of the car starting from B.

### When cars travel towards each other:
The relative speed of the cars when traveling towards each other is \( v_1 + v_2 \). Since they meet after \( T \) hours, we can write:
\[ (v_1 + v_2) \cdot T = 100 \]
\[ v_1 + v_2 = \frac{100}{T} \quad \text{(1)} \]

### When cars travel in the same direction:
The relative speed of the cars when traveling in the same direction is \( |v_1 - v_2| \). Since they meet after \( S \) hours, we can write:
\[ |v_1 - v_2| \cdot S = 100 \]
\[ |v_1 - v_2| = \frac{100}{S} \quad \text{(2)} \]

We have two equations now:
\[ v_1 + v_2 = \frac{100}{T} \quad \text{(1)} \]
\[ |v_1 - v_2| = \frac{100}{S} \quad \text{(2)} \]

### Solving these equations:
#### Case 1: \( v_1 \geq v_2 \)
In this case, \( v_1 - v_2 = \frac{100}{S} \).

From equation (1):
\[ v_1 = \frac{100}{T} - v_2 \]

Substitute this into equation (2):
\[ \left( \frac{100}{T} - v_2 \right) - v_2 = \frac{100}{S} \]
\[ \frac{100}{T} - 2v_2 = \frac{100}{S} \]
\[ 2v_2 = \frac{100}{T} - \frac{100}{S} \]
\[ v_2 = \frac{50}{T} - \frac{50}{S} \]

Now, substitute \( v_2 \) back into equation (1):
\[ v_1 + \left( \frac{50}{T} - \frac{50}{S} \right) = \frac{100}{T} \]
\[ v_1 = \frac{100}{T} - \frac{50}{T} + \frac{50}{S} \]
\[ v_1 = \frac{50}{T} + \frac{50}{S} \]

So, the speeds of the cars are:
\[ v_1 = \frac{50}{T} + \frac{50}{S} \]
\[ v_2 = \frac{50}{T} - \frac{50}{S} \]

#### Case 2: \( v_2 \geq v_1 \)
In this case, \( v_2 - v_1 = \frac{100}{S} \).

From equation (1):
\[ v_2 = \frac{100}{T} - v_1 \]

Substitute this into equation (2):
\[ v_2 - \left( \frac{100}{T} - v_1 \right) = \frac{100}{S} \]
\[ v_2 = \frac{50}{T} + \frac{50}{S} \]
\[ v_1 = \frac{50}{T} - \frac{50}{S} \]

So, the speeds of the cars are the same in both cases.

### Summary
The speeds of the cars are given by:
\[ v_1 = \frac{50}{T} + \frac{50}{S} \]
\[ v_2 = \frac{50}{T} - \frac{50}{S} \]

If you have any questions or need further details, please let me know.

Here are some related questions:

1. How do you derive the relative speed of two objects moving in the same direction?
2. How do you derive the relative speed of two objects moving towards each other?
3. Can you explain why the absolute value is used in the second equation?
4. What happens if the speeds of the cars are equal?
5. How would the equations change if the distance was different from 100 km?
6. How can you check if the calculated speeds are reasonable?
7. Can these types of problems be solved graphically?
8. How do these equations apply to real-world scenarios like trains or planes?

**Tip:** Always verify your units and ensure they are consistent throughout your calculations to avoid errors.

Explore how to solve speed, distance, and time problems involving two cars traveling towards or in the same direction. Understand the concepts of relative speed and apply them to calculate speeds when cars meet after certain times. Suitable for high school students studying physics or mathematics.

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