A particle whose speed is `50ms^(-1)` moves along the line from `A(2,1)` to `B(9,25)`. Find its velocity vector in the from of `ahat(i)+bhat(j)`.

Explanation

To find the velocity vector of a particle moving from point A(2,1) to point B(9,25) with a speed of 50 m/s, we can follow these steps:

### Step 1: Determine the displacement vector
The displacement vector \( \vec{d} \) from point A to point B can be calculated using the formula:
\[ \vec{d} = \vec{r}_B - \vec{r}_A \]
Where:
- \( \vec{r}_B = 9 \hat{i} + 25 \hat{j} \)
- \( \vec{r}_A = 2 \hat{i} + 1 \hat{j} \)

Calculating the displacement:
\[ \vec{d} = (9 \hat{i} + 25 \hat{j}) - (2 \hat{i} + 1 \hat{j}) = (9 - 2) \hat{i} + (25 - 1) \hat{j} = 7 \hat{i} + 24 \hat{j} \]

### Step 2: Calculate the magnitude of the displacement
The magnitude of the displacement \( |\vec{d}| \) can be calculated using the Pythagorean theorem:
\[ |\vec{d}| = \sqrt{(7)^2 + (24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ m} \]

### Step 3: Calculate the time taken
Using the formula for speed:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \implies \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
Substituting the values:
\[ t = \frac{25 \text{ m}}{50 \text{ m/s}} = \frac{1}{2} \text{ s} \]

### Step 4: Calculate the velocity vector
The velocity vector \( \vec{v} \) can be calculated using the formula:
\[ \vec{v} = \frac{\vec{d}}{t} \]
Substituting the values:
\[ \vec{v} = \frac{(7 \hat{i} + 24 \hat{j})}{\frac{1}{2}} = (7 \hat{i} + 24 \hat{j}) \cdot 2 = 14 \hat{i} + 48 \hat{j} \]

### Final Answer
Thus, the velocity vector in the form \( a \hat{i} + b \hat{j} \) is:
\[ \vec{v} = 14 \hat{i} + 48 \hat{j} \]

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