During inelastic collision between two bodies, which of the following quantities always remain conserved ?

Options

A. Total kinetic energy
B. Total mechanical energy
C. Total linear momentum
D. Speed of each body

Explanation

To solve the question regarding which quantity remains conserved during an inelastic collision between two bodies, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Inelastic Collision**:
- In an inelastic collision, two bodies collide and stick together or deform, resulting in a loss of kinetic energy.

2. **Identifying Conserved Quantities**:
- We need to determine which physical quantities are conserved during this type of collision. The options given are:
- Option 1: Total kinetic energy
- Option 2: Total mechanical energy
- Option 3: Total linear momentum
- Option 4: Speed of each body

3. **Analyzing Each Option**:
- **Option 1: Total Kinetic Energy**:
- In an inelastic collision, kinetic energy is not conserved. Some of it is transformed into other forms of energy (like heat or deformation).

- **Option 2: Total Mechanical Energy**:
- Mechanical energy (which includes kinetic and potential energy) is also not conserved in inelastic collisions due to energy transformation.

- **Option 3: Total Linear Momentum**:
- Linear momentum is conserved in an isolated system (where no external forces act). Since there are no external forces acting on the two bodies during the collision, the total linear momentum before the collision equals the total linear momentum after the collision.
- Mathematically, this can be expressed as:
\[ m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \]
- Here, \(m_1\) and \(m_2\) are the masses of the two bodies, and \(u_1\), \(u_2\), \(v_1\), and \(v_2\) are their initial and final velocities, respectively.

- **Option 4: Speed of Each Body**:
- The speeds of the bodies change during the collision, so this quantity is not conserved.

4. **Conclusion**:
- The only quantity that remains conserved during an inelastic collision is the total linear momentum. Therefore, the correct answer is **Option 3: Total Linear Momentum**.

### Final Answer:
The quantity that always remains conserved during an inelastic collision between two bodies is **Total Linear Momentum**.

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