A particle has an initial velocity of `4 hati +3 hatj` and an acceleration of `0.4 hati + 0.3 hatj`. Its speed after 10s is

Options

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

Explanation

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

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

### Step 2: Use the Formula for Final Velocity
To find the final velocity \( \mathbf{v} \) after a time \( t \), we use the formula:
\[ \mathbf{v} = \mathbf{u} + \mathbf{a} \cdot t \]
Substituting the values:
\[ t = 10 \, \text{s} \]
\[ \mathbf{v} = (4 \hat{i} + 3 \hat{j}) + (0.4 \hat{i} + 0.3 \hat{j}) \cdot 10 \]

### Step 3: Calculate the Contribution of Acceleration
Calculating \( \mathbf{a} \cdot t \):
\[ \mathbf{a} \cdot t = (0.4 \hat{i} + 0.3 \hat{j}) \cdot 10 = (4 \hat{i} + 3 \hat{j}) \]

### Step 4: Add Initial Velocity and Acceleration Contribution
Now we can add the initial velocity and the contribution from acceleration:
\[ \mathbf{v} = (4 \hat{i} + 3 \hat{j}) + (4 \hat{i} + 3 \hat{j}) = (8 \hat{i} + 6 \hat{j}) \]

### Step 5: Calculate the Magnitude of the Final Velocity
The magnitude of the final velocity \( |\mathbf{v}| \) is given by:
\[ |\mathbf{v}| = \sqrt{(8)^2 + (6)^2} \]
Calculating the squares:
\[ |\mathbf{v}| = \sqrt{64 + 36} = \sqrt{100} \]

### Step 6: Find the Final Speed
Thus, the speed after 10 seconds is:
\[ |\mathbf{v}| = 10 \, \text{units} \]

### Conclusion
The speed of the particle after 10 seconds is **10 units**.

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