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. 10 units
B. ` 7sqrt2 ` units
C. 7 units
D. 8.5 units

Explanation

To solve the problem step by step, we will follow the kinematic equations of motion.

### Step 1: Identify the given quantities
- 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 first equation of motion to find the final velocity
The first equation of motion states:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} t \]
Substituting the known values:
\[ \mathbf{v} = (3 \hat{i} + 4 \hat{j}) + (0.4 \hat{i} + 0.3 \hat{j}) \cdot 10 \]

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

### Step 4: Add the initial velocity and the acceleration component
Now, substituting back into the equation for final velocity:
\[ \mathbf{v} = (3 \hat{i} + 4 \hat{j}) + (4 \hat{i} + 3 \hat{j}) = (3 + 4) \hat{i} + (4 + 3) \hat{j} \]
\[ \mathbf{v} = 7 \hat{i} + 7 \hat{j} \]

### Step 5: Find the magnitude of the final velocity to determine speed
The magnitude of the velocity vector \( \mathbf{v} \) 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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