Problem 391 - JEE Physics

Question

A bag of sand of mass \(\mathrm{m}\) is suspended by rope. a bullet of mass \((\mathrm{m} / 30)\) is fired at it with a velocity \(\mathrm{V}\) and gets embedded into it. The velocity of the bag finally is (A) \((31 \mathrm{~V} / 30)\) (B) \((30 \mathrm{~V} / 31)\) (C) \((\mathrm{V} / 31)\) (D) \((\mathrm{V} / 30)\)

Step-by-Step Solution

Answer

The final velocity of the bag (\(V_f\)) after being hit by the bullet is: \(V_f = \frac{31V}{30}\). The correct answer is (A) \(\frac{31V}{30}\).

Step 1: Identify the initial momentum of the system

Before the collision, only the bullet is moving with a velocity of V, and the bag is stationary. So, the initial momentum of the system is the momentum of the bullet alone: Initial Momentum \(= m_{bullet} \times V\) Since the bullet mass is given as \(\frac{m}{30}\), we have: Initial Momentum \(= \frac{m}{30} \times V\)

Step 2: Identify the final momentum of the system

After the collision, the bullet gets embedded into the bag, and they move together with the same final velocity, which we can denote as \(V_f\). The total mass of the system after the collision is \(\frac{31m}{30}\). Therefore, the final momentum of the system will be: Final Momentum \(= (m + \frac{m}{30}) \times V_f = \frac{31m}{30} \times V_f\)

Step 3: The principle of conservation of linear momentum

According to the principle of conservation of linear momentum, the initial momentum of the system should equal the final momentum of the system. Therefore, we can set the initial momentum and final momentum equal to each other and solve for the final velocity, \(V_f\): \(\frac{m}{30} \times V = \frac{31m}{30} \times V_f\)

Step 4: Solve for the final velocity, \(V_f\)

Divide both sides of the equation by \(\frac{m}{30}\) to solve for the final velocity: \(V_f = \frac{(\frac{31m}{30} \times V_f)}{(\frac{m}{30} \times V)}\) Now, cancel out the terms \(\frac{m}{30}\) on both sides: \(V_f = \frac{31}{30} \times V\) Thus, the final velocity of the bag is: \(V_f = \frac{31V}{30}\) Comparing this result with the given options, we find that the correct answer is: (A) \(\frac{31V}{30}\)

Key Concepts

Inelastic Collision

In physics, an inelastic collision is a type of collision where the colliding objects stick together after the impact. In our exercise, the bullet embeds itself into the bag of sand, exemplifying perfect inelasticity. This means:

• The bullet and the bag move together as one mass after the collision.
• Some kinetic energy is lost in the process. Unlike elastic collisions, inelastic collisions do not conserve kinetic energy.
• The combined system has a new velocity after the collision.

In these scenarios, the primary focus is on conserving total momentum rather than energy.

Momentum

Momentum in physics is the product of an object's mass and its velocity, expressed as: ul>

Momentum (p) = Mass (m) × Velocity (v)

It is a vector quantity, meaning it has both magnitude and direction. The principle of conservation of momentum states that the total momentum of a closed system remains constant if no external forces act on it.
In our exercise, the bullet, before the collision, carries all of the system's momentum because it is the only moving object. Post-collision, this momentum is transferred to the combined mass of the bullet and the sandbag, resulting in a final momentum that still reflects the initial conditions.

Physics Problem Solving

When approaching a physics problem, it's crucial to break it down into manageable steps. For inelastic collisions, such as in this exercise:

• First, identify the initial state, determining all momentum contributors before the collision.
• Next, establish the final state, noting changes in momentum and ensuring conservation.
• Apply relevant physical principles, like momentum conservation, to link these states.
• Finally, solve algebraically for unknowns like final velocity, ensuring units are consistent throughout.

Sequentially addressing each component makes complex problems more approachable and solvable.

Final Velocity Calculation

Calculating the final velocity after an inelastic collision involves setting up an equation based on the conservation of momentum. For our scenario:

• We composed an equation using initial and final momentums, set equal to each other due to momentum conservation.
• The initial momentum is given by the bullet's mass and velocity, while the combined mass and final velocity dictate the final momentum.
• Solving for the final velocity involves rearranging and simplifying the equation, considering the increased mass post-collision.

Following these steps allows you to determine the final velocity, an essential value in understanding how the system behaves after the collision.