JEE Main 2021, 1 Sep Shift-II
LEVELJEE Main
Visualized Solution
The Sigma Insight: Work Done by Forces
The Braking Wagon: A Masterclass in the Work-Energy Theorem
The Runaway Wagon
Imagine a massive wagon cruising down a track at . Suddenly, the brakes are applied! The wagon doesn't just stop instantly; it grinds against the tracks, and a heavy-duty shock absorber (a spring) at the front compresses to help bring this mechanical beast to a halt.
This problem is a beautiful demonstration of how energy transforms and dissipates in the real world. Before we dive into the deep physics, we must ensure our mathematical language is consistent.
We start by converting the velocity into standard SI units. Multiplying by the conversion factor , we find that the wagon is moving at a brisk .
The Power of the Work-Energy Theorem
To understand how the wagon stops, we call upon one of the most elegant principles in physics: the Work-Energy Theorem.
This theorem states that the net work done on an object by all forces equals its change in kinetic energy.
In our scenario, the wagon eventually comes to a complete stop, meaning its final kinetic energy () is exactly zero. The initial kinetic energy () is the massive energy it possessed while moving at .
But what forces are doing the work? There are two main actors here: the friction from the brakes and the restoring force of the shock absorber's spring. Both of these forces act in the opposite direction to the wagon's motion, meaning they both do negative work, actively draining kinetic energy from the system.
The Energy Split
Friction and the Spring
The problem gives us a crucial piece of intel: of the wagon's kinetic energy is lost due to friction. This means the brakes are doing the heavy lifting.
If friction eats up of the energy, where does the remaining go? It must be absorbed by the spring! As the spring compresses by , it stores this remaining energy as elastic potential energy.
The work done by the spring is given by the classic formula:
Plugging these into our Work-Energy Theorem, we get a beautifully simple balance:
Rearranging this, we see exactly what we deduced logically: the energy stored in the spring equals of the initial kinetic energy.
The Final Crunch
Now, we substitute the raw physics into our refined equation. We know that .
Notice how the cancels out on both sides, leaving us with a straightforward calculation.
To match the format requested by the problem, we express this in scientific notation:
The value we are looking for is 16.
This problem perfectly illustrates how engineers design safety systems. By knowing how much energy the brakes can handle, they can calculate exactly how stiff the shock absorber needs to be to safely absorb the rest!
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