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

Options

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

Explanation

To solve the problem, we need to find the speed of a particle after 10 seconds given its initial velocity and acceleration. Let's break down the solution step by step.

### Step 1: Identify the Given Information
- Initial velocity \( \mathbf{u} = 3 \hat{i} + 4 \hat{j} \) m/s
- Acceleration \( \mathbf{a} = 0.4 \hat{i} + 0.3 \hat{j} \) m/s²
- Time \( t = 10 \) s

### Step 2: Use the Equation of Motion
We can use the equation of motion to find the final velocity \( \mathbf{v} \):
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} \cdot t \]

### Step 3: Substitute the Values
Substituting the 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
Now calculate \( \mathbf{a} \cdot t \):
\[ \mathbf{a} \cdot t = (0.4 \hat{i} + 0.3 \hat{j}) \cdot 10 = (0.4 \cdot 10) \hat{i} + (0.3 \cdot 10) \hat{j} = 4 \hat{i} + 3 \hat{j} \]

### Step 5: Add the Initial Velocity and the Acceleration Component
Now add this result to the initial velocity:
\[ \mathbf{v} = (3 \hat{i} + 4 \hat{j}) + (4 \hat{i} + 3 \hat{j}) = (3 + 4) \hat{i} + (4 + 3) \hat{j} = 7 \hat{i} + 7 \hat{j} \]

### Step 6: Find the Magnitude of the Final Velocity
The magnitude of the final velocity (which is the speed) is given by:
\[ |\mathbf{v}| = \sqrt{(7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7 \sqrt{2} \]

### Final Answer
The speed of the particle after 10 seconds is \( 7 \sqrt{2} \) m/s.

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