Two particles of equal mass have coordinates (2m,4m,6m) and (6m,2m,8m). Of these one particle has a velocity `v_(1) = (2i)ms^(-1)` and another particle has a velocity `v_(2) = (2j)ms^(-1)` at time t=0. The coordinate of their center of mass at time t=1s will be

Options

A. (4m,4m,7m)
B. (5m,4m,7m)
C. (2m,4m,6m)
D. (4m,5m,4m)

Explanation

To find the coordinates of the center of mass of the two particles at time \( t = 1 \) second, we will follow these steps:

### Step 1: Identify the initial coordinates and velocities of the particles
- Particle A has initial coordinates \( (x_1, y_1, z_1) = (2 \, \text{m}, 4 \, \text{m}, 6 \, \text{m}) \) and velocity \( \vec{v_1} = (2 \hat{i}) \, \text{m/s} \).
- Particle B has initial coordinates \( (x_2, y_2, z_2) = (6 \, \text{m}, 2 \, \text{m}, 8 \, \text{m}) \) and velocity \( \vec{v_2} = (2 \hat{j}) \, \text{m/s} \).

### Step 2: Calculate the new coordinates of each particle after 1 second
- For Particle A (moving in the x-direction):
\[ x_{1, \text{new}} = x_1 + v_{1,x} \cdot t = 2 \, \text{m} + (2 \, \text{m/s} \cdot 1 \, \text{s}) = 4 \, \text{m} \]
\[ y_{1, \text{new}} = y_1 = 4 \, \text{m} \quad (\text{no change in y}) \]
\[ z_{1, \text{new}} = z_1 = 6 \, \text{m} \quad (\text{no change in z}) \]
Therefore, the new coordinates of Particle A are \( (4 \, \text{m}, 4 \, \text{m}, 6 \, \text{m}) \).

- For Particle B (moving in the y-direction):
\[ x_{2, \text{new}} = x_2 = 6 \, \text{m} \quad (\text{no change in x}) \]
\[ y_{2, \text{new}} = y_2 + v_{2,y} \cdot t = 2 \, \text{m} + (2 \, \text{m/s} \cdot 1 \, \text{s}) = 4 \, \text{m} \]
\[ z_{2, \text{new}} = z_2 = 8 \, \text{m} \quad (\text{no change in z}) \]
Therefore, the new coordinates of Particle B are \( (6 \, \text{m}, 4 \, \text{m}, 8 \, \text{m}) \).

### Step 3: Calculate the center of mass coordinates
The center of mass \( (x_{cm}, y_{cm}, z_{cm}) \) for two particles of equal mass \( m \) is given by:
\[ x_{cm} = \frac{m x_1 + m x_2}{m + m} = \frac{x_1 + x_2}{2} \]
\[ y_{cm} = \frac{m y_1 + m y_2}{m + m} = \frac{y_1 + y_2}{2} \]
\[ z_{cm} = \frac{m z_1 + m z_2}{m + m} = \frac{z_1 + z_2}{2} \]

Substituting the new coordinates:
- For \( x_{cm} \):
\[ x_{cm} = \frac{4 \, \text{m} + 6 \, \text{m}}{2} = \frac{10 \, \text{m}}{2} = 5 \, \text{m} \]

- For \( y_{cm} \):
\[ y_{cm} = \frac{4 \, \text{m} + 4 \, \text{m}}{2} = \frac{8 \, \text{m}}{2} = 4 \, \text{m} \]

- For \( z_{cm} \):
\[ z_{cm} = \frac{6 \, \text{m} + 8 \, \text{m}}{2} = \frac{14 \, \text{m}}{2} = 7 \, \text{m} \]

### Final Result
The coordinates of the center of mass at \( t = 1 \, \text{s} \) are:
\[ (x_{cm}, y_{cm}, z_{cm}) = (5 \, \text{m}, 4 \, \text{m}, 7 \, \text{m}) \]