Sigma Percentile
JEE Main 2021, 27 Aug Shift-II
LEVELJEE Main

Animated Solution for Physics - Kinematics: Water drops are falling from a nozzle of a shower onto the floor, from a height of . The drops fall at a regular interval of time. When the first drop strikes the floor, at that instant, the third drop begins to fall. Locate the position of second drop from the floor when the first drop strikes the floor.

Select Answer:

Visualized Solution

  • Let be the height of the nozzle from the floor.
  • The drops fall from rest, so initial velocity .

  • Drops fall at regular intervals .
  • (Time for 1st drop)
  • (Time for 2nd drop)

  • Using the second equation of motion for the first drop:

  • Time taken by the second drop:

  • Distance fallen by the second drop from the nozzle:

  • Position of the second drop from the floor:

  • Always read carefully: 'distance from top' vs 'position from floor'.
  • What if the 4th drop was just starting? How would the time intervals change?

The Sigma Insight: Motion in a Straight Line

Solution Diagram

The Rhythm of Falling Drops

Have you ever watched a leaky faucet or a showerhead and noticed the mesmerizing, rhythmic pattern of the falling drops? This problem takes that everyday observation and turns it into a beautiful exercise in kinematics.
The core trick to solving this problem isn't a complex formula; it's understanding the timeline of events. The problem states that the drops fall at regular intervals. When the first drop hits the floor, the third drop is just starting its journey.

Mapping the Time Intervals

Imagine a stopwatch that clicks every time a drop leaves the nozzle. Let's call this interval .
When drop 1 leaves, the time is . When drop 2 leaves, the time is . When drop 3 leaves, the time is .
The problem tells us that at , the first drop hits the ground. This means the total time of flight for the first drop, let's call it , is exactly .
At this exact moment, how long has the second drop been falling? It started at , so it has been falling for a duration of .
This gives us our master relationship:

The Master Equation

Now, we can use the second equation of motion to find the total time . Since the drops fall from rest, the initial velocity .
Substituting our values for the first drop:
Notice how elegantly the cancels out on both sides!

Finding the Second Drop

Since the second drop has been falling for half the time, its time of flight is:
Let's find out how far it has fallen from the nozzle. We'll call this distance .

The Final Trap

Here is where many students lose marks. They see , spot it in the options, and confidently tick it. But wait! The question specifically asks for the position of the second drop from the floor, not from the nozzle.
To find the height from the floor, we subtract the distance fallen from the total height:
And there we have it! By carefully mapping the timeline and avoiding the final trap, we arrive at the correct answer.

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