A particle of mass `m_(1) = 4 kg` moving at `6hatims^(-1)` collides perfectly elastically with a particle of mass `m_(2) = 2` moving at `3hati ms^(-1)`

Options

A. Velocity of centre of mass (CM) is `5hatims^(-1)`
B. The velocities of the particles relative to the centre of mass have same magnitude.
C. Speed of individual particle before and after collision remains same.
D. The velocity of particles relative to CM after collision are `vecv_(1f//cm)=-hatims^-1,vecv_(2f//cm)=2hatims^(-1)`

Explanation

To solve the problem step by step, we will analyze the given information about the two particles, apply the relevant formulas, and check each statement provided.

### Given:
- Mass of particle 1, \( m_1 = 4 \, \text{kg} \)
- Initial velocity of particle 1, \( \vec{v_1} = 6 \hat{i} \, \text{m/s} \)
- Mass of particle 2, \( m_2 = 2 \, \text{kg} \)
- Initial velocity of particle 2, \( \vec{v_2} = 3 \hat{i} \, \text{m/s} \)

### Step 1: Calculate the velocity of the center of mass (COM)

The velocity of the center of mass (\( \vec{V}_{cm} \)) is given by the formula:

\[ \vec{V}_{cm} = \frac{m_1 \vec{v_1} + m_2 \vec{v_2}}{m_1 + m_2} \]

Substituting the values:

\[ \vec{V}_{cm} = \frac{(4 \, \text{kg} \cdot 6 \hat{i} \, \text{m/s}) + (2 \, \text{kg} \cdot 3 \hat{i} \, \text{m/s})}{4 \, \text{kg} + 2 \, \text{kg}} \]

Calculating the numerator:

\[ = \frac{24 \hat{i} + 6 \hat{i}}{6 \, \text{kg}} = \frac{30 \hat{i}}{6} = 5 \hat{i} \, \text{m/s} \]

### Step 2: Check Statement A

Statement A claims that the velocity of the center of mass is \( 5 \hat{i} \, \text{m/s} \). From our calculation, we found that this is true.

### Step 3: Calculate the relative velocities of the particles with respect to the center of mass

The relative velocity of particle 1 with respect to the center of mass is:

\[ \vec{v_{1,cm}} = \vec{v_1} - \vec{V}_{cm} = 6 \hat{i} - 5 \hat{i} = 1 \hat{i} \, \text{m/s} \]

The relative velocity of particle 2 with respect to the center of mass is:

\[ \vec{v_{2,cm}} = \vec{v_2} - \vec{V}_{cm} = 3 \hat{i} - 5 \hat{i} = -2 \hat{i} \, \text{m/s} \]

### Step 4: Check Statement B

Statement B claims that the velocities of the particles relative to the center of mass have the same magnitude. We found that:

- Magnitude of \( \vec{v_{1,cm}} = 1 \, \text{m/s} \)
- Magnitude of \( \vec{v_{2,cm}} = 2 \, \text{m/s} \)

Since the magnitudes are not the same, Statement B is false.

### Step 5: Calculate final velocities after the elastic collision

Using the formulas for perfectly elastic collisions, we can find the final velocities:

For particle 1:

\[ \vec{v_1'} = \frac{m_1 - m_2}{m_1 + m_2} \vec{v_1} + \frac{2 m_2}{m_1 + m_2} \vec{v_2} \]

Substituting the values:

\[ \vec{v_1'} = \frac{4 - 2}{4 + 2} \cdot 6 \hat{i} + \frac{2 \cdot 2}{6} \cdot 3 \hat{i} \]

Calculating:

\[ = \frac{2}{6} \cdot 6 \hat{i} + \frac{4}{6} \cdot 3 \hat{i} = 2 \hat{i} + 2 \hat{i} = 4 \hat{i} \, \text{m/s} \]

For particle 2:

\[ \vec{v_2'} = \frac{2 m_1}{m_1 + m_2} \vec{v_1} + \frac{m_2 - m_1}{m_1 + m_2} \vec{v_2} \]

Substituting the values:

\[ \vec{v_2'} = \frac{2 \cdot 4}{6} \cdot 6 \hat{i} + \frac{2 - 4}{6} \cdot 3 \hat{i} \]

Calculating:

\[ = \frac{8}{6} \cdot 6 \hat{i} - \frac{2}{6} \cdot 3 \hat{i} = 8 \hat{i} - 1 \hat{i} = 7 \hat{i} \, \text{m/s} \]

### Step 6: Check Statement C

Statement C claims that the speeds of individual particles before and after the collision remain the same. We found:

- Speed of particle 1 before = 6 m/s, after = 4 m/s
- Speed of particle 2 before = 3 m/s, after = 7 m/s

Since the speeds are not the same, Statement C is false.

### Step 7: Check Statement D

For Statement D, we need to find the relative velocities after the collision:

Relative velocity of particle 1 with respect to the center of mass after collision:

\[ \vec{v_{1,cm}}' = \vec{v_1'} - \vec{V}_{cm} = 4 \hat{i} - 5 \hat{i} = -1 \hat{i} \, \text{m/s} \]

Relative velocity of particle 2 with respect to the center of mass after collision:

\[ \vec{v_{2,cm}}' = \vec{v_2'} - \vec{V}_{cm} = 7 \hat{i} - 5 \hat{i} = 2 \hat{i} \, \text{m/s} \]

Thus, Statement D is true.

### Final Conclusion

- Statement A: True
- Statement B: False
- Statement C: False
- Statement D: True