Two point masses 1 and 2 move with uniform velocities `vec(v)_(1)` and `vec(v)_(2)`, respectively. Their initial position vectors are `vec(r )_(1)` and `vec(r )_(2)`, respectively. Which of the following should be satisfied for the collision of the point masses?

Options

A. `(vec(r )_(1)-vec(r )_(2))/(|vec(r )_(2)-vec(r )_(1)|)=(vec(v)_(2)-vec(v)_(1))/(|vec(v)_(2)-vec(v)_(1)|)`
B. `(vec(r )_(2)-vec(r )_(1))/(|vec(r )_(2)-vec(r )_(1)|)=(vec(v)_(2)-vec(v)_(1))/(|vec(v)_(2)-vec(v)_(1)|)`
C. `(vec(r)_(2)-vec(r )_(1))/(|vec(r )_(2)+vec(r )_(1)|)=(vec(v)_(2)-vec(v)_(1))/(|vec(v)_(2)+vec(v)_(1)|)`
D. `(vec(r)_(2)-vec(r )_(1))/(|vec(r )_(2)+vec(r )_(1)|)=(vec(v)_(2)-vec(v)_(1))/(|vec(v)_(2)+vec(v)_(1)|)`

Explanation

To determine the condition for the collision of two point masses moving with uniform velocities, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Initial Conditions**:
Let the initial position vectors of the two point masses be represented as:
- \( \vec{r}_1 \) for mass 1
- \( \vec{r}_2 \) for mass 2

Their velocities are given as:
- \( \vec{v}_1 \) for mass 1
- \( \vec{v}_2 \) for mass 2

2. **Write the Position Vector Equations**:
The position of each mass at time \( t \) can be expressed as:
- For mass 1: \( \vec{r}_1(t) = \vec{r}_1 + \vec{v}_1 t \)
- For mass 2: \( \vec{r}_2(t) = \vec{r}_2 + \vec{v}_2 t \)

3. **Set the Condition for Collision**:
For the two masses to collide, their position vectors must be equal at some time \( t \):
\[ \vec{r}_1 + \vec{v}_1 t = \vec{r}_2 + \vec{v}_2 t \]

4. **Rearranging the Equation**:
Rearranging the above equation gives:
\[ \vec{r}_1 - \vec{r}_2 = (\vec{v}_2 - \vec{v}_1) t \]

5. **Expressing in Terms of Vectors**:
We can express the left-hand side as:
\[ \vec{r}_1 - \vec{r}_2 = \vec{d} \quad \text{(where } \vec{d} \text{ is the vector from mass 2 to mass 1)} \]
And the right-hand side as:
\[ (\vec{v}_2 - \vec{v}_1) t \]

6. **Finding the Relationship**:
To find a relationship between the vectors, we can equate the directions:
\[ \frac{\vec{d}}{|\vec{d}|} = \frac{\vec{v}_2 - \vec{v}_1}{|\vec{v}_2 - \vec{v}_1|} \]

7. **Final Condition**:
Thus, the condition for collision can be summarized as:
\[ \frac{\vec{r}_1 - \vec{r}_2}{|\vec{r}_1 - \vec{r}_2|} = \frac{\vec{v}_2 - \vec{v}_1}{|\vec{v}_2 - \vec{v}_1|} \]