Sigma Percentile
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Animated Solution for Physics - Kinematics: In an announcement on a railway station, a passenger hears that the last train has passed the station min earlier than his train. On the next station that is km away from the previous station, in another announcement he hears that the first train arrived min earlier than his train. Reading time from his watch, he calculates average speed of his train to be km/h. Relying on the announcements and the passenger's calculations, determine average speed of the first train.

Visualized Solution

  • Let's plot a position-time () graph for the two trains.
  • Station A is at and Station B is at .

  • The passenger's train travels distance at speed .
  • Time taken .

  • Let the first train take time to travel from Station A to Station B.

  • At Station A, was ahead by .
  • At Station B, was ahead by .

  • From the timeline, the total time from 's departure to 's arrival can be written in two ways:

The Sigma Insight: Motion in a Straight Line

Solution Diagram

The Power of the Position-Time Graph

When dealing with relative motion and time gaps, equations can sometimes become a tangled mess of variables. However, visualizing the scenario using a position-time () graph turns a complex algebraic puzzle into a simple geometry problem.
Imagine the -axis representing time and the -axis representing the distance along the railway track. Station A is our origin at , and Station B is located at . We have two trains: the passenger's train () and the first train ().

Analyzing the Passenger's Journey

Let's start with what we know completely: the passenger's train. It covers a distance of at an average speed of . Using the fundamental kinematic relation, we can find the time it takes for this journey:
Converting this into minutes for easier comparison with our given data, we get . On our graph, the passenger's train is a straight line starting at some time and ending at , where the horizontal width of this line is exactly .

Connecting the Timelines

Now, let's introduce the first train, . We don't know its speed, so let's assume it takes a time to travel between the two stations.
The announcements give us the crucial links between the two trains. At Station A, passed earlier than . At Station B, arrived earlier than .
If we look at the total time elapsed from the moment leaves Station A to the moment arrives at Station B, we can trace it in two distinct paths along our graph's time axis: 1. Follow 's journey () and then add the final waiting gap at Station B (). 2. Add the initial waiting gap at Station A () and then follow 's journey ().
Equating these two paths gives us our master equation:

The Final Calculation

Rearranging the equation to solve for the unknown time , we get:
Substituting our known values:
So, the first train took , or , to cover the distance. Finally, we calculate its average speed :
The General Formula: If we substitute the symbolic expressions back into our final step, we can derive a beautiful general formula for any problem of this type:
This elegant result shows how the relative time gaps directly modulate the effective speed of the leading object.

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