Question

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

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

Answer 1

The Correct Option is C

Force, Velocity, and Power Calculation 

We are given the force as a function of time and need to calculate the velocity and power of the object:

Step 1: Given Force

The force acting on the object is given by:

\(\vec{F} = 2t \hat{i} + 3 t^2 \hat{j}\)

Step 2: Using Newton's Second Law of Motion

According to Newton's second law, the force is related to the rate of change of velocity:

\(m \frac{d \vec{v}}{dt} = 2 t \hat{i} + 3 t^2 \hat{j}\)

Where: - \( m = 1 \, \text{kg} \) (mass of the object), - \( \vec{v} \) is the velocity of the object, - \( t \) is time.

Step 3: Finding the Velocity

We now integrate the equation with respect to time to find the velocity. The equation becomes:

\(\int\limits^{\hat{v}}_0 d \vec{v} = \int\limits^t_0 (2t \hat{i} + 3 t^2 \hat{j}) \, dt\)

Performing the integration, we get:

\(\vec{v} = t^2 \hat{i} + t^3 \hat{j}\)

This gives the velocity of the object as a function of time:

\(\vec{v} = t^2 \hat{i} + t^3 \hat{j}\)

Step 4: Calculating the Power

The power \( P \) delivered by the force is the dot product of the force and velocity vectors:

\(P = \vec{F} \cdot \vec{v}\)

Substitute the expressions for \( \vec{F} \) and \( \vec{v} \):

\(P = (2 t \hat{i} + 3 t^2 \hat{j}) \cdot (t^2 \hat{i} + t^3 \hat{j})\)

Now compute the dot product:

\(P = (2 t^3 + 3 t^5) \, \text{W}\)

Conclusion:

The power delivered by the force as a function of time is:

\(P = 2 t^3 + 3 t^5 \, \text{W}\)