A bullet in motion hits and gets embedded in a solid block resting on a frictionless table. What is conserved ?

Options

A. Momentum and KE
B. Kinetic energy alone
C. Neither KE nor momentum
D. Momentum alone

Answer

To solve the question regarding the conservation laws during the collision of a bullet with a block on a frictionless table, let's break it down step by step.

### Step 1: Understand the Scenario
We have a bullet that is moving and collides with a solid block that is resting on a frictionless table. The bullet gets embedded in the block after the collision.

**Hint:** Identify the objects involved and their initial states (motion vs. rest).

### Step 2: Identify the Type of Collision
Since the bullet gets embedded in the block, this is classified as a perfectly inelastic collision. In this type of collision, the two objects stick together after the impact.

**Hint:** Remember that in a perfectly inelastic collision, the two objects move together as one mass after the collision.

### Step 3: Conservation of Momentum
In any collision, momentum is always conserved. The total momentum before the collision must equal the total momentum after the collision.

- Let the mass of the bullet be \( m_1 \) and its initial velocity be \( u_1 \).
- Let the mass of the block be \( m_2 \) and its initial velocity be \( u_2 = 0 \) (since it is at rest).
- After the collision, both the bullet and block move together with a common velocity \( v \).

Using the conservation of momentum:
\[ m_1 u_1 + m_2 u_2 = (m_1 + m_2) v \]
Since \( u_2 = 0 \):
\[ m_1 u_1 = (m_1 + m_2) v \]

**Hint:** Write down the momentum conservation equation to analyze the relationship between the initial and final states.

### Step 4: Kinetic Energy Consideration
In a perfectly inelastic collision, kinetic energy is not conserved. Some kinetic energy is transformed into other forms of energy (like heat, sound, etc.) during the collision.

- Initial kinetic energy of the system:
\[ KE_{initial} = \frac{1}{2} m_1 u_1^2 \]
- Final kinetic energy of the system after the collision:
\[ KE_{final} = \frac{1}{2} (m_1 + m_2) v^2 \]

Since \( v \) is derived from the momentum conservation equation, it will generally be less than the initial kinetic energy of the bullet alone, indicating a loss of kinetic energy.

**Hint:** Compare the initial and final kinetic energies to conclude that kinetic energy is not conserved.

### Step 5: Conclusion
From the analysis, we conclude that during the collision:
- **Momentum is conserved.**
- **Kinetic energy is not conserved.**

Thus, the correct answer to the question "What is conserved?" is **momentum alone**.

**Final Answer:** Momentum alone is conserved.