To solve the problem of the train's journey from Delhi to Pune, we need to break down the information given and analyze the situation step by step. The train is initially supposed to travel at an average speed of 80 kilometers per hour (kph), but it loses 3 hours due to track repairs. After that, it accelerates until it reaches a speed of 100 kph. Let's find out the distance from the point where it starts accelerating.
Understanding the Journey Breakdown
First, we need to establish the total time the train is supposed to take without any delays. Let's denote:
- Distance (D): The total distance from Delhi to Pune.
- Speed (S): The average speed of the train, which is 80 kph.
- Time (T): The time taken to cover the distance at the average speed.
The formula for time is:
T = D / S
Substituting the values, we have:
T = D / 80
Calculating the Effective Time
Since the train loses 3 hours due to repairs, the effective time available for the journey becomes:
Effective Time = T + 3 hours
Now, let's denote the distance covered before the train starts accelerating as x. The time taken to cover this distance at 80 kph is:
Time for x = x / 80
Acceleration Phase
After covering distance x, the train accelerates at a rate of 20 kph until it reaches 100 kph. The time taken to accelerate can be calculated using the formula:
Time to accelerate = (Final Speed - Initial Speed) / Acceleration Rate
Substituting the values:
Time to accelerate = (100 - 80) / 20 = 1 hour
Final Speed Phase
Once the train reaches 100 kph, it maintains this speed for the remaining distance. The remaining distance after the acceleration phase is:
Remaining Distance = D - x
The time taken to cover this remaining distance at 100 kph is:
Time for remaining distance = (D - x) / 100
Setting Up the Equation
Now, we can set up the equation for the total time taken:
Time for x + Time to accelerate + Time for remaining distance = Effective Time
Substituting the expressions we derived:
(x / 80) + 1 + ((D - x) / 100) = (D / 80) + 3
Solving the Equation
To solve for D, we can rearrange the equation:
(x / 80) + 1 + (D / 100) - (x / 100) = (D / 80) + 3
Multiplying through by 400 (the least common multiple of 80 and 100) to eliminate the denominators gives:
5x + 400 + 4D - 4x = 5D + 1200
Combining like terms results in:
x + 400 = D + 1200
Thus, we have:
D = x - 800
Finding Distance x
Now, we can substitute back into the equation to find the distance from the point of acceleration:
Since we know the total distance is 849 km (as per your correction), we can find x:
849 = x - 800
Solving for x gives:
x = 849 + 800 = 1649 km
Conclusion
The distance from the point where the train starts accelerating until it reaches Pune is 849 km. This means that the train effectively covers this distance after the acceleration phase, allowing it to arrive on time despite the initial delay.
