Problem 4 - Master Resource Book in JEE Main Physics

Question

A bullet fired from a gun with a velocity of \(10^{4} \mathrm{~ms}^{-1}\) goes through a bag full of straw. If the bullet loses half of its kinetic energy in the bag, its velocity when it comes out of the bag will be (a) \(7071.06 \mathrm{~ms}^{-1}\) (b) \(707 \mathrm{~ms}^{-1}\) (c) \(70.71 \mathrm{~ms}^{-1}\) (d) \(707.06 \mathrm{~ms}^{-1}\)

Step-by-Step Solution

Answer

The velocity of the bullet when it comes out of the bag is \(7071.06 \mathrm{~ms}^{-1}\).

Step 1: Understand the Problem

We are given the initial velocity of the bullet, which is \(10^4 \mathrm{~ms}^{-1}\). The bullet loses half of its kinetic energy as it passes through the straw bag. We need to determine the bullet's velocity after it exits the bag.

Step 2: Recall the Kinetic Energy Formula

The formula for kinetic energy (KE) of an object with mass \(m\) and velocity \(v\) is given by \( KE = \frac{1}{2} m v^2 \).

Step 3: Set Up the Equation for Kinetic Energy Loss

If the bullet loses half of its kinetic energy, the kinetic energy after exiting the bag (\(KE_{\text{final}}\)) is half of the initial kinetic energy (\(KE_{\text{initial}}\)). Therefore, \[ KE_{\text{final}} = \frac{1}{2} KE_{\text{initial}}. \]

Step 4: Express Velocities in Terms of Kinetic Energies

Substitute the formula for kinetic energy into the equation: \[ \frac{1}{2} m v_{\text{final}}^2 = \frac{1}{2} \left( \frac{1}{2} m v_{\text{initial}}^2 \right). \] Simplifying, we get: \[ v_{\text{final}}^2 = \frac{1}{2} v_{\text{initial}}^2. \]

Step 5: Solve for Final Velocity

From the equation \( v_{\text{final}}^2 = \frac{1}{2} v_{\text{initial}}^2 \), take the square root of both sides to find \( v_{\text{final}} \): \[ v_{\text{final}} = \sqrt{ \frac{1}{2} } \times v_{\text{initial}}. \] This simplifies to: \[ v_{\text{final}} = \frac{v_{\text{initial}}}{\sqrt{2}}. \]

Step 6: Substitute Initial Velocity and Calculate

Substitute \( v_{\text{initial}} = 10^4 \mathrm{~ms}^{-1} \) into the equation: \[ v_{\text{final}} = \frac{10^4}{\sqrt{2}} \approx 7071.06 \mathrm{~ms}^{-1}. \]

Key Concepts

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is a concept from physics that is crucial when discussing moving objects. For any object, including a bullet, the kinetic energy can be calculated using the formula: \[ KE = \frac{1}{2} m v^2 \] where

• \( KE \) is the kinetic energy,
• \( m \) is the mass of the object, and
• \( v \) is the velocity of the object.

This formula shows that the kinetic energy depends directly on the mass and the square of the velocity. This means if an object's velocity doubles, its kinetic energy increases by four times. Similarly, changes in mass also directly affect kinetic energy. Understanding kinetic energy helps in solving physics problems, especially those that involve changes in velocity as seen with the bullet in our exercise.

Velocity Calculation

Velocity calculation is a key part of solving problems that involve movement, such as the bullet passing through the straw bag. Velocity is the speed of an object in a given direction, and it can change when an object interacts with forces or other substances. Given that the bullet loses half of its kinetic energy, we need to find its new velocity after losing this energy. By manipulating the kinetic energy formula, we established that after the bullet loses half its kinetic energy, its new velocity is given by:\[ v_{\text{final}} = \frac{v_{\text{initial}}}{\sqrt{2}} \]Here,

• \( v_{\text{final}} \) is the final velocity of the bullet after the energy loss,
• \( v_{\text{initial}} \) is its initial velocity before entering the straw bag.

In our exercise, substituting the initial velocity of \( 10^4 \, \text{ms}^{-1} \), we calculate a final velocity of approximately \( 7071.06 \, \text{ms}^{-1} \). This calculation shows how velocities change considering energy transformations without needing to measure mass.

Conservation of Energy

The principle of conservation of energy is a fundamental concept in physics. It states that energy cannot be created or destroyed in an isolated system. It can only be transformed from one form to another. In the context of our exercise, the principle explains how the bullet's kinetic energy reduces as it passes through the straw bag. The energy the bullet loses is not gone; it has transformed, likely into heat due to friction between the bullet and the straw. We use conservation of energy to understand how the bullet has half its kinetic energy when exiting the straw, as any transformation of energy is accounted for by reducing its velocity. This principle helps solidify why changes in motion or other states within a system always boil down to energy transformation, maintaining the total energy of the system.

Newtonian Mechanics

Newtonian mechanics, based on Isaac Newton's laws of motion, serves as the foundation for understanding motion and forces. It helps us describe how objects move and interact with each other. For the bullet example, Newton's second law, \( F = ma \) (force equals mass times acceleration), implies that the interaction between the bullet and the straw affects the bullet's motion. As the bullet encounters the resistance of the straw, it experiences a force opposite its motion, slowing it down.Additionally, Newton's third law, which states that for every action there is an equal and opposite reaction, explains why the energy loss occurs — the straw exerts a reaction force on the bullet. Using Newtonian mechanics allows us to quantify these interactions and understand the resulting changes in velocity and energy. Its concepts are crucial for analyzing any physical movement, especially involving significant forces like those between a speeding bullet and resistance offering materials.