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

Answer

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

### Step 1: Identify the initial velocity and acceleration
The initial velocity \( \mathbf{u} \) is given as:
\[ \mathbf{u} = 3 \hat{i} + 4 \hat{j} \]
The acceleration \( \mathbf{a} \) is given as:
\[ \mathbf{a} = 0.4 \hat{i} + 0.3 \hat{j} \]

### Step 2: Use the formula for final velocity
The formula for the final velocity \( \mathbf{v} \) after time \( t \) is:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} \cdot t \]
Here, \( t = 10 \, \text{s} \).

### Step 3: Calculate the change in velocity due to acceleration
First, 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 4: Add the initial velocity to the change in velocity
Now, add this 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 5: Calculate the speed (magnitude of the velocity)
The speed is the magnitude of the velocity vector \( \mathbf{v} \):
\[ |\mathbf{v}| = \sqrt{(7)^2 + (7)^2} = \sqrt{49 + 49} = \sqrt{98} = 7\sqrt{2} \]

### Conclusion
The speed of the particle after 10 seconds is:
\[ 7\sqrt{2} \, \text{units} \]

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