Two particles P and Q are moving with velocity of `(hat(i)+hat(j))` and `(-hat(i)+2hat(j))` respectively. At time `t=0`, P is at origin and Q is at a point with positive vector `(2hat(i)+hat(j))`. Then the shortest distance between P & Q is :-

Options

A. `(2sqrt(5))/5`
B. `(4sqrt(5))/5`
C. `sqrt(5)`
D. `(3sqrt(5))/5`

Explanation

To find the shortest distance between two particles P and Q moving with given velocities, we can follow these steps:

### Step 1: Understand the position and velocity of particles
- Particle P has a velocity vector **vP** = \( \hat{i} + \hat{j} \).
- Particle Q has a velocity vector **vQ** = \( -\hat{i} + 2\hat{j} \).
- At time \( t = 0 \), particle P is at the origin (0,0) and particle Q is at the position vector \( \mathbf{r_Q} = 2\hat{i} + \hat{j} \).

### Step 2: Write the position vectors as functions of time
- The position of particle P at time \( t \) is:
\[ \mathbf{r_P}(t) = \hat{i}t + \hat{j}t = t(\hat{i} + \hat{j}) \]
- The position of particle Q at time \( t \) is:
\[ \mathbf{r_Q}(t) = (2 - t)\hat{i} + (1 + 2t)\hat{j} \]

### Step 3: Find the distance between P and Q
- The distance \( d(t) \) between P and Q at time \( t \) can be expressed as:
\[ d(t) = |\mathbf{r_Q}(t) - \mathbf{r_P}(t)| \]
- Substituting the position vectors:
\[ d(t) = |[(2 - t) - t]\hat{i} + [(1 + 2t) - t]\hat{j}| \]
\[ = |(2 - 2t)\hat{i} + (1 + t)\hat{j}| \]

### Step 4: Calculate the magnitude of the distance
- The magnitude of the distance vector is:
\[ d(t) = \sqrt{(2 - 2t)^2 + (1 + t)^2} \]
\[ = \sqrt{(2 - 2t)^2 + (1 + t)^2} \]
\[ = \sqrt{(4 - 8t + 4t^2) + (1 + 2t + t^2)} \]
\[ = \sqrt{5t^2 - 6t + 5} \]

### Step 5: Minimize the distance function
- To find the minimum distance, we can differentiate \( d(t)^2 \) with respect to \( t \) and set it to zero:
\[ f(t) = 5t^2 - 6t + 5 \]
\[ f'(t) = 10t - 6 \]
Setting \( f'(t) = 0 \):
\[ 10t - 6 = 0 \implies t = \frac{3}{5} \]

### Step 6: Calculate the minimum distance
- Substitute \( t = \frac{3}{5} \) back into the distance formula:
\[ d\left(\frac{3}{5}\right) = \sqrt{5\left(\frac{3}{5}\right)^2 - 6\left(\frac{3}{5}\right) + 5} \]
\[ = \sqrt{5 \cdot \frac{9}{25} - \frac{18}{5} + 5} \]
\[ = \sqrt{\frac{45}{25} - \frac{90}{25} + \frac{125}{25}} = \sqrt{\frac{80}{25}} = \sqrt{3.2} = \frac{4}{\sqrt{5}} \]

### Final Answer
The shortest distance between particles P and Q is:
\[ \frac{4}{\sqrt{5}} \]

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