Consider the following two statements:
A. Linear momentum of a system of particles is zero.
B. Kinetic energy of a system of particles is zero.

Options

A. A implies B and B implies A.
B. A does not imply B and B does not imply A.
C. A implies B but B does not imply A.
D. B implies A but A does not imply B.

Answer

To analyze the two statements regarding linear momentum and kinetic energy of a system of particles, we will go through the definitions and implications step by step.

### Step 1: Understand the Definitions
- **Linear Momentum (p)** of a system of particles is defined as the vector sum of the momenta of all the individual particles. Mathematically, it is given by:
\[ p = \sum_{i=1}^{n} m_i v_i \]
where \( m_i \) is the mass and \( v_i \) is the velocity of the \( i^{th} \) particle.

- **Kinetic Energy (KE)** of a system of particles is defined as the sum of the kinetic energies of all the individual particles. It is expressed as:
\[ KE = \sum_{i=1}^{n} \frac{1}{2} m_i v_i^2 \]

### Step 2: Analyze Statement A
- **Statement A**: Linear momentum of a system of particles is zero.
- This means:
\[ \sum_{i=1}^{n} m_i v_i = 0 \]
This can occur if the velocities of the particles are such that they cancel each other out (e.g., one particle moving in one direction and another moving in the opposite direction with equal momentum).

### Step 3: Analyze Statement B
- **Statement B**: Kinetic energy of a system of particles is zero.
- This implies:
\[ \sum_{i=1}^{n} \frac{1}{2} m_i v_i^2 = 0 \]
Since mass \( m_i \) is always positive, the only way for the kinetic energy to be zero is if all velocities \( v_i \) are zero. Therefore, if the kinetic energy is zero, all particles must be at rest.

### Step 4: Determine Implications
- **Does A imply B?**
- If linear momentum is zero, it does not necessarily mean that kinetic energy is zero. For example, consider two particles of equal mass moving in opposite directions with equal speed. Their momenta cancel out, resulting in zero linear momentum, but they still possess kinetic energy.

- **Does B imply A?**
- If kinetic energy is zero, then all particles must have zero velocity, which means their momenta are also zero. Thus, if kinetic energy is zero, linear momentum must also be zero.

### Conclusion
From our analysis:
- **B implies A**: If the kinetic energy of a system is zero, then the linear momentum is also zero.
- **A does not imply B**: If the linear momentum is zero, the kinetic energy can still be greater than zero.

### Final Answer
Thus, the correct conclusion is that B implies A, but A does not imply B.