To determine the minimum time a train can take to travel a distance \( d \) with the greatest acceleration or deceleration \( a \), we can use the principles of kinematics. The question essentially revolves around the relationship between distance, acceleration, and time. Let's break it down step by step.
Understanding the Basics of Motion
When an object accelerates from rest, its motion can be described using the following kinematic equation:
d = (1/2) a t²
In this equation, \( d \) is the distance traveled, \( a \) is the acceleration, and \( t \) is the time taken to cover that distance. If we rearrange this equation to solve for time \( t \), we get:
t = √(2d/a)
Applying the Formula
Now, let’s apply this formula to our scenario. Since the train can also decelerate, we can assume that it accelerates to a maximum speed and then decelerates to stop at the next station. The total time taken for this journey will be the time taken to accelerate to maximum speed plus the time taken to decelerate.
However, if we consider the minimum time to travel the distance \( d \) under constant acceleration, we can simplify our calculations. The train will accelerate for half the distance and then decelerate for the other half. Thus, the total distance can be split into two equal parts:
- First half: \( d/2 \) during acceleration
- Second half: \( d/2 \) during deceleration
Calculating Time for Each Half
Using the formula for each half, we find the time taken to accelerate to the midpoint:
t₁ = √(2(d/2)/a) = √(d/a)
Similarly, the time taken to decelerate from the maximum speed back to rest is the same:
t₂ = √(2(d/2)/a) = √(d/a)
Finding Total Time
The total time \( T \) for the entire journey is the sum of the two times:
T = t₁ + t₂ = √(d/a) + √(d/a) = 2√(d/a)
Conclusion
From our calculations, we can conclude that the minimum time \( T \) for the train to travel a distance \( d \) with maximum acceleration \( a \) is:
T = 2√(d/a)
Therefore, the correct option from the choices provided is:
c) 2(d/a)^(1/2)
This result illustrates the fundamental principles of motion under constant acceleration and how they can be applied to real-world scenarios like train travel.
