Two particles A and B are placed in gravity free space at `(0, 0, 0) m and (30, 0, 0) m` respectively. Particle A is projected with a velocity `(5hat i + 10 hat j + 5 hat k) ms^-1,` while particle B is projected with a velocity `(10 hat i + 5 hat j + 5 hat k) ms^-1` simultaneously. Then,

Options

A. they will collide at `(10, 20, 10) m`
B. they will collide at `(10, 10, 10) m`
C. they will never collide
D. they will collide at 2 s

Explanation

To solve the problem of whether particles A and B will collide, we need to analyze their positions over time based on their initial positions and velocities.

### Step-by-Step Solution:

1. **Identify Initial Positions and Velocities:**
- Particle A is at position \( \mathbf{r_A} = (0, 0, 0) \, \text{m} \) and has a velocity \( \mathbf{v_A} = (5 \hat{i} + 10 \hat{j} + 5 \hat{k}) \, \text{m/s} \).
- Particle B is at position \( \mathbf{r_B} = (30, 0, 0) \, \text{m} \) and has a velocity \( \mathbf{v_B} = (10 \hat{i} + 5 \hat{j} + 5 \hat{k}) \, \text{m/s} \).

2. **Write the Position Equations:**
- The position of particle A at time \( t \) is given by:
\[ \mathbf{r_A}(t) = \mathbf{r_{A0}} + \mathbf{v_A} t = (0, 0, 0) + (5 \hat{i} + 10 \hat{j} + 5 \hat{k}) t = (5t, 10t, 5t) \]
- The position of particle B at time \( t \) is given by:
\[ \mathbf{r_B}(t) = \mathbf{r_{B0}} + \mathbf{v_B} t = (30, 0, 0) + (10 \hat{i} + 5 \hat{j} + 5 \hat{k}) t = (30 + 10t, 5t, 5t) \]

3. **Set the Position Equations Equal for Collision:**
- For the particles to collide, their positions must be equal:
\[ \mathbf{r_A}(t) = \mathbf{r_B}(t) \]
- This gives us three equations:
1. \( 5t = 30 + 10t \) (for the x-coordinate)
2. \( 10t = 5t \) (for the y-coordinate)
3. \( 5t = 5t \) (for the z-coordinate)

4. **Solve the Equations:**
- From the first equation:
\[ 5t - 10t = 30 \implies -5t = 30 \implies t = -6 \, \text{s} \]
- The second equation simplifies to:
\[ 10t - 5t = 0 \implies 5t = 0 \implies t = 0 \, \text{s} \]
- The third equation is always true for any \( t \).

5. **Analyze the Results:**
- The first equation gives a negative time, which is not physically possible. This indicates that the particles will never collide.
- The second equation shows that they are at the same position at \( t = 0 \), but they will diverge from that point due to their different velocities.

### Conclusion:
Since the time calculated for collision is negative, it confirms that particles A and B will never collide.