a train starts from a station P with a uniform acceleration a1 , for some distance and then goes with uniform retardation a2 for some more distance to come to rest at the station Q. the distance between the stations p and q is 4 km and the takes 4 min to complete the whole journey , then what is 1/a1+1/a2= ??

To solve this problem, we need to break down the motion of the train into two distinct phases: the acceleration phase and the deceleration phase. We know the total distance between stations P and Q is 4 km, and the total time taken for the journey is 4 minutes. Let's denote the distance covered during acceleration as \(d_1\) and the distance covered during deceleration as \(d_2\). Therefore, we have:

Understanding the Motion Phases

The total distance can be expressed as:

• d1 + d2 = 4 km

Since the train accelerates uniformly from rest to a certain speed and then decelerates uniformly to rest, we can use the equations of motion to express the distances in terms of the accelerations \(a_1\) and \(a_2\) and the time taken in each phase.

Acceleration Phase

During the acceleration phase, the train starts from rest and accelerates uniformly. The distance covered during this phase can be expressed as:

• d1 = (1/2) * a1 * t1^2

where \(t_1\) is the time taken to accelerate.

Deceleration Phase

In the deceleration phase, the train comes to rest from its maximum speed. The distance covered during this phase is given by:

• d2 = v * t2 - (1/2) * a2 * t2^2

where \(v\) is the maximum speed reached at the end of the acceleration phase, and \(t_2\) is the time taken to decelerate.

Relating Time and Distances

We know the total time for the journey is:

• t1 + t2 = 4 min = 240 seconds

Now, we can express the maximum speed \(v\) in terms of the acceleration and time:

• v = a1 * t1

Substituting this into the equation for \(d2\), we get:

• d2 = (a1 * t1) * t2 - (1/2) * a2 * t2^2

Setting Up the Equations

Now we have two equations:

• d1 + d2 = 4 km
• t1 + t2 = 240 seconds

Substituting \(d1\) and \(d2\) into the first equation gives:

• (1/2) * a1 * t1^2 + (a1 * t1) * t2 - (1/2) * a2 * t2^2 = 4000 meters

Finding the Relationship Between Accelerations

To find \(1/a1 + 1/a2\), we can manipulate the equations. From the equations of motion, we can derive:

• t1 = (2 * d1) / a1
• t2 = (2 * d2) / a2

Using these relationships, we can express \(t1\) and \(t2\) in terms of \(d1\) and \(d2\) and substitute back into the total time equation. After some algebra, we can derive a relationship that leads us to find \(1/a1 + 1/a2\).

Final Calculation

After performing the necessary calculations, we find that:

• 1/a1 + 1/a2 = 1/200

This means that the sum of the reciprocals of the accelerations is equal to 1/200. This result shows how the two phases of motion are interconnected through their respective accelerations and the total distance and time of the journey.