A body of mass `1kg ` begins to move under the action of a time dependent force `vec(F)=(2t hat(i)+3t^(2) hat(j))N`, where `hat (i) ` and `hat(j)` are unit vectors along `X` and `Y` axis. What does will be developed by the source at time `t` ?

Options

A. `(2t^(2)+4r^(4))W`
B. `(2t^(2)+3r^(4))W`
C. `(2t^(3)+3r^(5))W`
D. `(2t+3r^(3))W`

Explanation

To solve the problem, we need to determine the power developed by a body of mass 1 kg under the influence of a time-dependent force given by \(\vec{F} = (2t \hat{i} + 3t^2 \hat{j}) \, \text{N}\).

### Step-by-Step Solution:

1. **Identify the Force and Mass**:
The force acting on the body is:
\[ \vec{F} = (2t \hat{i} + 3t^2 \hat{j}) \, \text{N} \]
The mass of the body is:
\[ m = 1 \, \text{kg} \]

2. **Use Newton's Second Law**:
According to Newton's second law, the force is equal to mass times acceleration:
\[ \vec{F} = m \vec{a} \]
Since \(m = 1 \, \text{kg}\), we have:
\[ \vec{a} = \vec{F} = (2t \hat{i} + 3t^2 \hat{j}) \, \text{m/s}^2 \]

3. **Find the Velocity**:
The acceleration is the time derivative of velocity:
\[ \vec{a} = \frac{d\vec{v}}{dt} \]
Thus, we can write:
\[ \frac{d\vec{v}}{dt} = (2t \hat{i} + 3t^2 \hat{j}) \]
Integrating both sides with respect to time \(t\):
\[ \vec{v} = \int (2t \hat{i} + 3t^2 \hat{j}) \, dt \]
Performing the integration:
\[ \vec{v} = \left( t^2 \hat{i} + t^3 \hat{j} \right) + \vec{C} \]
Assuming the initial velocity is zero (the body starts from rest), we have:
\[ \vec{C} = 0 \quad \Rightarrow \quad \vec{v} = t^2 \hat{i} + t^3 \hat{j} \]

4. **Calculate Power**:
Power \(P\) developed by the force is given by the dot product of the force and velocity:
\[ P = \vec{F} \cdot \vec{v} \]
Substituting the expressions for \(\vec{F}\) and \(\vec{v}\):
\[ P = (2t \hat{i} + 3t^2 \hat{j}) \cdot (t^2 \hat{i} + t^3 \hat{j}) \]
Calculating the dot product:
\[ P = 2t \cdot t^2 + 3t^2 \cdot t^3 = 2t^3 + 3t^5 \]

5. **Final Result**:
Therefore, the power developed by the body at time \(t\) is:
\[ P(t) = 2t^3 + 3t^5 \, \text{W} \]