Calculating the Car's Speed in an Overtaking Scenario

This problem involves calculating the speed of a car based on its relative position and movement compared to a bus over a specific time period. We need to determine the car's speed using the given initial distance, final distance, time elapsed, and the bus's speed.

Understanding the Problem

• The bus is initially 60 km ahead of the car.
• Both are moving in the same direction on a straight road.
• After 3 hours, the car is 90 km ahead of the bus.
• The speed of the bus ($S_B$) is given as 45 km/h.
• We need to find the speed of the car ($S_C$).

Step-by-Step Solution

To solve this, we can look at the change in the relative distance between the car and the bus. Initially, the bus is ahead. Finally, the car is ahead. This means the car has closed the initial gap and created a new gap in its favor.

1. Initial Relative Position:

The bus is ahead of the car by 60 km. We can represent this as:

Initial distance of Bus ahead of Car = $60$ km

2. Final Relative Position:

After 3 hours, the car is ahead of the bus by 90 km. We can represent this as:

Final distance of Car ahead of Bus = $90$ km

3. Total Change in Relative Distance:

The total change in the relative position is the sum of the initial gap the car needed to cover and the new gap the car created ahead of the bus.

Total relative distance covered by the car with respect to the bus = Initial gap + Final gap

Total relative distance = $60 \text{ km} + 90 \text{ km} = 150 \text{ km}$

4. Time Taken:

The time elapsed for this change in relative distance is given as 3 hours.

Time ($t$) = $3$ hours

5. Calculating Relative Speed:

The relative speed is the rate at which the distance between the two vehicles changes. Since they are moving in the same direction, the relative speed of the car with respect to the bus is $S_C - S_B$.

Relative Speed = $\frac{\text{Total relative distance}}{\text{Time}}$

Relative Speed = $\frac{150 \text{ km}}{3 \text{ hours}} = 50 \text{ km/h}$

6. Finding the Car's Speed:

We know that the relative speed ($S_{relative}$) is the difference between the car's speed ($S_C$) and the bus's speed ($S_B$).

$S_{relative} = S_C - S_B$

We calculated the relative speed as 50 km/h, and we are given the bus speed $S_B = 45$ km/h.

$50 \text{ km/h} = S_C - 45 \text{ km/h}$

Now, we solve for $S_C$:

$S_C = 50 \text{ km/h} + 45 \text{ km/h}$

$S_C = 95 \text{ km/h}$

Conclusion

The speed of the car is 95 km/h.