JEE Main 2021, 18 March Shift-I
LEVELJEE Main
Visualized Solution
The Sigma Insight: Newton's Laws of Motion
The Physics of a Bullet Piercing Wood
Imagine a high-speed bullet tearing through the air and slamming into a solid wooden block. The wood doesn't just let the bullet pass freely; it fights back! It exerts a massive resistive force, rapidly draining the bullet's kinetic energy until it comes to a dead stop.
This classic physics problem is a beautiful demonstration of how kinematics and dynamics work together. Let's break down the journey of this bullet and uncover the exact force the wood uses to stop it.
Analyzing the Setup
Before we dive into the equations, let's lay out what we know. We have a bullet with a mass of . It strikes the wooden block with an initial velocity of .
The wood acts like a brake, applying a uniform deceleration. Eventually, the bullet comes to a complete halt, meaning its final velocity is . We are also told that the bullet penetrates a distance of into the block before stopping.
Crucial Step: In physics, consistency in units is everything. Since our velocity is in meters per second, we must convert the penetration distance into meters. Therefore, our displacement is .
The Kinematics of Deceleration
To find the force exerted by the wood, we first need to understand how fast the bullet was slowing down. This is where kinematics comes to the rescue. Since the deceleration is uniform, we can use the third equation of motion, which perfectly links velocity, acceleration, and displacement without needing to know the time:
Let's substitute our known values into this elegant equation:
Now, we perform the atomic computation. The left side becomes , and on the right side, simplifies to exactly .
The negative sign here is not a mistake; it is a profound physical statement. It tells us that the acceleration is acting in the opposite direction to the bullet's motion, confirming that the bullet is indeed experiencing retardation.
Newton's Second Law in Action
With the acceleration in hand, we have unlocked the door to finding the force. We call upon Sir Isaac Newton's legendary Second Law of Motion, which states that the net force acting on an object is the product of its mass and its acceleration:
We simply plug in the mass of the bullet and the acceleration we just discovered:
The force exerted by the wooden block is . Again, the negative sign indicates that this is a retarding force, pushing against the bullet's forward motion.
The question specifically asks for the magnitude of this effective retarding force. The magnitude is simply the absolute value, stripping away the directional sign.
Therefore, the magnitude of the force is , making our final answer .
The Work-Energy Perspective
As a quick thought experiment, could we have solved this differently? Absolutely! The beauty of physics is that different fundamental principles often lead to the same truth.
We could have used the Work-Energy Theorem, which states that the work done by the net force equals the change in kinetic energy:
If you plug the values into this energy equation, you will arrive at the exact same elegant result of . Whether you view the universe through the lens of forces and acceleration or work and energy, the laws of physics remain beautifully consistent!
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