A force `F = 2hati + 4hatj` Newton displaces the body by `s = 3hatj + 5hatk` meter in 2s. The power generated will be

Options

A. 11 W
B. 6 W
C. 22 W
D. 12 W

Answer

To find the power generated by the force acting on the body, we can follow these steps:

### Step 1: Identify the given quantities
- Force \( \mathbf{F} = 2\hat{i} + 4\hat{j} \) Newton
- Displacement \( \mathbf{s} = 3\hat{j} + 5\hat{k} \) meters
- Time \( t = 2 \) seconds

### Step 2: Calculate the velocity
The velocity \( \mathbf{v} \) can be calculated using the formula:
\[ \mathbf{v} = \frac{\mathbf{s}}{t} \]
Substituting the values:
\[ \mathbf{v} = \frac{3\hat{j} + 5\hat{k}}{2} = \frac{3}{2}\hat{j} + \frac{5}{2}\hat{k} \]

### Step 3: Calculate the power
Power \( P \) is given by the dot product of the force and velocity:
\[ P = \mathbf{F} \cdot \mathbf{v} \]
Substituting the values of \( \mathbf{F} \) and \( \mathbf{v} \):
\[ P = (2\hat{i} + 4\hat{j}) \cdot \left(\frac{3}{2}\hat{j} + \frac{5}{2}\hat{k}\right) \]

### Step 4: Compute the dot product
Calculating the dot product:
\[ P = 2\hat{i} \cdot \frac{3}{2}\hat{j} + 4\hat{j} \cdot \frac{3}{2}\hat{j} + 2\hat{i} \cdot \frac{5}{2}\hat{k} + 4\hat{j} \cdot \frac{5}{2}\hat{k} \]
Since \( \hat{i} \cdot \hat{j} = 0 \) and \( \hat{i} \cdot \hat{k} = 0 \):
\[ P = 0 + 4 \cdot \frac{3}{2} + 0 + 0 = 6 \text{ Watts} \]

### Final Answer
The power generated is \( 6 \) Watts.

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