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 \( t = 1 \, \text{s} \), we can follow these steps:

### Step 1: Determine the Initial Coordinates of the Particles
The coordinates of the two particles are given as:
- Particle 1: \( (x_1, y_1, z_1) = (2 \, \text{m}, 4 \, \text{m}, 6 \, \text{m}) \)
- Particle 2: \( (x_2, y_2, z_2) = (6 \, \text{m}, 2 \, \text{m}, 8 \, \text{m}) \)

### Step 2: Calculate the Center of Mass at \( t = 0 \)
The formula for 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 values:
\[ x_{cm} = \frac{2 + 6}{2} = \frac{8}{2} = 4 \, \text{m} \]
\[ y_{cm} = \frac{4 + 2}{2} = \frac{6}{2} = 3 \, \text{m} \]
\[ z_{cm} = \frac{6 + 8}{2} = \frac{14}{2} = 7 \, \text{m} \]

Thus, the initial center of mass is at \( (4 \, \text{m}, 3 \, \text{m}, 7 \, \text{m}) \).

### Step 3: Determine the Velocities of the Particles
The velocities of the particles are given as:
- Particle 1: \( \mathbf{v_1} = (2 \, \mathbf{i}) \, \text{m/s} \)
- Particle 2: \( \mathbf{v_2} = (2 \, \mathbf{j}) \, \text{m/s} \)

### Step 4: Calculate the Displacement of Each Particle After 1 Second
The displacement of each particle after \( t = 1 \, \text{s} \) can be calculated as:
- Displacement of Particle 1: \( \mathbf{d_1} = \mathbf{v_1} \cdot t = (2 \, \mathbf{i}) \cdot 1 = (2 \, \text{m}, 0, 0) \)
- Displacement of Particle 2: \( \mathbf{d_2} = \mathbf{v_2} \cdot t = (0, 2 \, \mathbf{j}) \cdot 1 = (0, 2 \, \text{m}, 0) \)

### Step 5: Calculate the New Coordinates of Each Particle
Now, we can find the new coordinates after 1 second:
- New coordinates of Particle 1:
\[ (x_1', y_1', z_1') = (2 + 2, 4 + 0, 6 + 0) = (4 \, \text{m}, 4 \, \text{m}, 6 \, \text{m}) \]
- New coordinates of Particle 2:
\[ (x_2', y_2', z_2') = (6 + 0, 2 + 2, 8 + 0) = (6 \, \text{m}, 4 \, \text{m}, 8 \, \text{m}) \]

### Step 6: Calculate the New Center of Mass at \( t = 1 \, \text{s} \)
Now we can calculate the new center of mass using the new coordinates:
\[ x_{cm}' = \frac{4 + 6}{2} = \frac{10}{2} = 5 \, \text{m} \]
\[ y_{cm}' = \frac{4 + 4}{2} = \frac{8}{2} = 4 \, \text{m} \]
\[ z_{cm}' = \frac{6 + 8}{2} = \frac{14}{2} = 7 \, \text{m} \]

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