A particle of mass `1 kg` has velocity `vec(v)_(1) = (2t)hat(i)` and another particle of mass `2 kg` has velocity `vec(v)_(2) = (t^(2))hat(j)`
`{:(,"Column I",,"Column II",),((A),"Netforce on centre of mass at 2 s",(p),(20)/(9) unit,),((B),"Velocity of centre of mass at 2 s",(q),sqrt68 " unit",),((C ),"Displacement of centre of mass in 2s",(r ),(sqrt80)/(3) "unit",),(,,(s),"None",):}`

Explanation

To solve the problem step-by-step, we will analyze the motion of two particles and calculate the required quantities: net force on the center of mass, velocity of the center of mass, and displacement of the center of mass.

### Step 1: Calculate the acceleration of each particle
1. **For Particle 1 (mass = 1 kg)**:
- Given velocity: \(\vec{v}_1 = (2t) \hat{i}\)
- Acceleration \(\vec{a}_1 = \frac{d\vec{v}_1}{dt} = \frac{d}{dt}(2t \hat{i}) = 2 \hat{i}\) (m/s²)

2. **For Particle 2 (mass = 2 kg)**:
- Given velocity: \(\vec{v}_2 = (t^2) \hat{j}\)
- Acceleration \(\vec{a}_2 = \frac{d\vec{v}_2}{dt} = \frac{d}{dt}(t^2 \hat{j}) = 2t \hat{j}\) (m/s²)

### Step 2: Calculate the acceleration of the center of mass
- Total mass \(M = m_1 + m_2 = 1 \, \text{kg} + 2 \, \text{kg} = 3 \, \text{kg}\)
- Acceleration of the center of mass \(\vec{a}_{CM}\) is given by:
\[ \vec{a}_{CM} = \frac{m_1 \vec{a}_1 + m_2 \vec{a}_2}{M} \]
Substituting the values:
\[ \vec{a}_{CM} = \frac{1 \cdot (2 \hat{i}) + 2 \cdot (2t \hat{j})}{3} = \frac{2 \hat{i} + 4t \hat{j}}{3} = \frac{2}{3} \hat{i} + \frac{4t}{3} \hat{j} \]

### Step 3: Calculate the net force on the center of mass at \(t = 2 \, \text{s}\)
- Substitute \(t = 2\):
\[ \vec{a}_{CM} = \frac{2}{3} \hat{i} + \frac{8}{3} \hat{j} \]
- Net force \(\vec{F}_{CM} = M \vec{a}_{CM}\):
\[ \vec{F}_{CM} = 3 \left(\frac{2}{3} \hat{i} + \frac{8}{3} \hat{j}\right) = 2 \hat{i} + 8 \hat{j} \]

### Step 4: Calculate the magnitude of the net force
\[ |\vec{F}_{CM}| = \sqrt{(2)^2 + (8)^2} = \sqrt{4 + 64} = \sqrt{68} \, \text{N} \]

### Step 5: Calculate the velocity of the center of mass at \(t = 2 \, \text{s}\)
- Velocity of the center of mass \(\vec{v}_{CM}\) is given by:
\[ \vec{v}_{CM} = \frac{m_1 \vec{v}_1 + m_2 \vec{v}_2}{M} \]
Substituting the values:
- For \(t = 2\):
\[ \vec{v}_1 = 4 \hat{i}, \quad \vec{v}_2 = 4 \hat{j} \]
\[ \vec{v}_{CM} = \frac{1 \cdot (4 \hat{i}) + 2 \cdot (4 \hat{j})}{3} = \frac{4 \hat{i} + 8 \hat{j}}{3} = \frac{4}{3} \hat{i} + \frac{8}{3} \hat{j} \]

### Step 6: Calculate the magnitude of the velocity of the center of mass
\[ |\vec{v}_{CM}| = \sqrt{\left(\frac{4}{3}\right)^2 + \left(\frac{8}{3}\right)^2} = \sqrt{\frac{16}{9} + \frac{64}{9}} = \sqrt{\frac{80}{9}} = \frac{\sqrt{80}}{3} \, \text{m/s} \]

### Step 7: Calculate the displacement of the center of mass in 2 seconds
- Displacement \(\vec{s}_{CM}\) can be calculated by integrating the velocity:
\[ \vec{s}_{CM} = \int_0^2 \vec{v}_{CM} \, dt \]
Using the expression for \(\vec{v}_{CM}\):
\[ \vec{s}_{CM} = \int_0^2 \left(\frac{2}{3} t \hat{i} + \frac{t^2}{3} \hat{j}\right) dt \]
Calculating the integrals:
\[ \vec{s}_{CM} = \left[\frac{2}{3} \cdot \frac{t^2}{2} \hat{i} + \frac{1}{3} \cdot \frac{t^3}{3} \hat{j}\right]_0^2 = \left[\frac{2}{3} \cdot 2^2 \hat{i} + \frac{1}{3} \cdot \frac{8}{3} \hat{j}\right] = \left[\frac{8}{3} \hat{i} + \frac{8}{9} \hat{j}\right] \]

### Step 8: Calculate the magnitude of the displacement
\[ |\vec{s}_{CM}| = \sqrt{\left(\frac{8}{3}\right)^2 + \left(\frac{8}{9}\right)^2} = \sqrt{\frac{64}{9} + \frac{64}{81}} = \sqrt{\frac{576 + 64}{81}} = \sqrt{\frac{640}{81}} = \frac{8\sqrt{10}}{9} \]

### Final Answers
- **Net force on center of mass at 2 s**: \(\sqrt{68} \, \text{N}\)
- **Velocity of center of mass at 2 s**: \(\frac{\sqrt{80}}{3} \, \text{m/s}\)
- **Displacement of center of mass in 2s**: \(\frac{20}{9} \, \text{units}\)