A particle has initial velocity `(3 hat i+ 4 hat j)` and has acceleration `(0.4 hat i+ 0.3 hat j)`. Its speed after `10 s` is.

Options

A. 7 unit
B. `7sqrt(2)`
C. 8.5 unit
D. 10 unit

Answer

To find the speed of the particle after 10 seconds, we can follow these steps:

### Step 1: Identify the given values
- Initial velocity, \( \mathbf{u} = 3 \hat{i} + 4 \hat{j} \)
- Acceleration, \( \mathbf{a} = 0.4 \hat{i} + 0.3 \hat{j} \)
- Time, \( t = 10 \, s \)

### Step 2: Use the formula for final velocity
The final velocity \( \mathbf{v} \) can be calculated using the equation:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} \cdot t \]

### Step 3: Substitute the values into the equation
Substituting the known values into the equation:
\[ \mathbf{v} = (3 \hat{i} + 4 \hat{j}) + (0.4 \hat{i} + 0.3 \hat{j}) \cdot 10 \]

### Step 4: Calculate the acceleration component
Calculate the acceleration component multiplied by time:
\[ \mathbf{a} \cdot t = (0.4 \hat{i} + 0.3 \hat{j}) \cdot 10 = (4 \hat{i} + 3 \hat{j}) \]

### Step 5: Add the initial velocity and the acceleration component
Now add the initial velocity and the calculated acceleration:
\[ \mathbf{v} = (3 \hat{i} + 4 \hat{j}) + (4 \hat{i} + 3 \hat{j}) = (3 + 4) \hat{i} + (4 + 3) \hat{j} = 7 \hat{i} + 7 \hat{j} \]

### Step 6: Calculate the magnitude of the final velocity
The magnitude of the final velocity \( |\mathbf{v}| \) is given by:
\[ |\mathbf{v}| = \sqrt{(v_x)^2 + (v_y)^2} = \sqrt{(7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \]

### Final Answer
Thus, the speed of the particle after 10 seconds is:
\[ 7\sqrt{2} \]

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