A bus went from Isfahan to Tehran at a speed of 80 km per hour. Two hours after the car, it started on the same route at a speed of 100 km per hour. Calculate how many hours does the car arrive at the bus?

To solve this problem, we will use relative motion concepts and set up equations based on the information provided.

### Step-by-Step Solution:

1. **Let \( t \) be the time (in hours) it takes for the car to catch up with the bus** after it starts the journey.

2. **The bus travels at 80 km/h**, and since it starts two hours earlier, by the time the car starts, the bus has already traveled:

   \[
   \text{Distance traveled by bus in 2 hours} = 80 \times 2 = 160 \, \text{km}.
   \]

3. **The car travels at 100 km/h**. After the car starts, the bus is still moving at 80 km/h. The car needs to cover the 160 km gap plus catch up to the bus's ongoing distance.

4. **Relative speed**: Since both vehicles are moving in the same direction, the relative speed between the car and the bus is:

   \[
   \text{Relative speed} = 100 - 80 = 20 \, \text{km/h}.
   \]

5. **Time for the car to catch up**: To close the 160 km gap, the car needs to travel at the relative speed of 20 km/h:

   \[
   \text{Time} = \frac{\text{Distance}}{\text{Relative Speed}} = \frac{160}{20} = 8 \, \text{hours}.
   \]

Thus, **it will take the car 8 hours** to catch up to the bus after it starts.

### Final Answer:
The car will catch up to the bus **8 hours** after it begins its journey.

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Would you like further details, or do you have any questions? Here are some related questions to deepen your understanding:

1. What if the car's speed was only 90 km/h instead of 100 km/h? How long would it take then?
2. How far would the car travel before catching up with the bus?
3. How far would the bus have traveled by the time the car catches up?
4. What if the bus had started 3 hours earlier instead of 2 hours earlier? How would that change the answer?
5. If the distance between Isfahan and Tehran is 400 km, would the car catch up before reaching Tehran?

**Tip**: When two objects are moving in the same direction, their relative speed is the difference between their individual speeds.

Learn how to calculate the time it takes for a car traveling at 100 km/h to catch up to a bus traveling at 80 km/h with a 2-hour head start. This step-by-step solution covers concepts like relative speed and distance-time relationships. Suitable for students in Grades 8-10.

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