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

Options

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

Explanation

To solve the problem step by step, we will follow these steps:

### Step 1: Identify the given information
We have:
- Mass of the body, \( m = 1 \, \text{kg} \)
- Force acting on the body, \( \vec{F} = (2t \hat{i} + 3t^2 \hat{j}) \, \text{N} \)

### Step 2: Write the relationship between force, mass, and acceleration
According to Newton's second law, the force acting on an object is equal to the mass of the object multiplied by its acceleration:
\[ \vec{F} = m \vec{a} \]
Since \( m = 1 \, \text{kg} \), we have:
\[ \vec{F} = \vec{a} \]

### Step 3: Relate acceleration to velocity
Acceleration can be expressed as the time derivative of velocity:
\[ \vec{a} = \frac{d\vec{v}}{dt} \]
Thus, we can write:
\[ \vec{F} = \frac{d\vec{v}}{dt} \]

### Step 4: Set up the equation for velocity
From the force equation, we have:
\[ \frac{d\vec{v}}{dt} = 2t \hat{i} + 3t^2 \hat{j} \]
To find the velocity, we integrate both sides with respect to time \( t \):
\[ d\vec{v} = (2t \hat{i} + 3t^2 \hat{j}) dt \]
Integrating:
\[ \vec{v} = \int (2t \hat{i} + 3t^2 \hat{j}) dt = (t^2 \hat{i} + t^3 \hat{j}) + \vec{C} \]
Assuming the initial velocity is zero, we can ignore the constant of integration \( \vec{C} \):
\[ \vec{v} = t^2 \hat{i} + t^3 \hat{j} \]

### Step 5: Calculate power developed by the force
Power \( P \) developed by the force is given by the dot product of 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:
\[ P = (2t \cdot t^2) + (3t^2 \cdot t^3) = 2t^3 + 3t^5 \]

### Step 6: Final expression for power
Thus, the power developed by the force at time \( t \) is:
\[ P = 2t^3 + 3t^5 \, \text{W} \]

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