The Question is from neet-physics work-energy-power , and it was created by the author VENKATESWARARAO

Question

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

VENKATESWARARAO · Accepted Answer

## CORE INFORMATION
✓ **Answer**: D
## CONCEPT FRAMEWORK
	**Primary Concept**: Power as the dot product of force and velocity
	**Related Topics**: Newton's Second Law, Integration, Vector Calculus
	**Prerequisites**: Understanding of force, velocity, power, and basic calculus.
## SOLUTION PATHWAY
**Given Information**:
	Force: $\vec{F} = (2t\hat{i} + 3t^2\hat{j})$ N
	Mass: m = 1 kg

**Method & Steps**:
	1. Find the acceleration using Newton's Second Law.
		Reasoning: Newton's Second Law states that $\vec{F} = m\vec{a}$. Since mass is 1 kg, acceleration is equal to the force.
		Formula/Rule: $\vec{a} = \frac{\vec{F}}{m}$
	2. Find the velocity by integrating the acceleration with respect to time.
		Working: $\vec{a} = (2t\hat{i} + 3t^2\hat{j})$
		$\vec{v} = \int \vec{a} dt = \int (2t\hat{i} + 3t^2\hat{j}) dt = (t^2\hat{i} + t^3\hat{j})$
		Key Point: We assume the initial velocity to be zero.
	3. Calculate the power using the dot product of force and velocity.
		Working: $P = \vec{F} \cdot \vec{v} = (2t\hat{i} + 3t^2\hat{j}) \cdot (t^2\hat{i} + t^3\hat{j})$
		$P = (2t)(t^2) + (3t^2)(t^3) = 2t^3 + 3t^5$
		Result: $P = 2t^3 + 3t^5$
		Verification: The units are consistent, and the answer aligns with the expected form.
## OPTION ANALYSIS
[A] :x: Incorrect calculation.
[B] :x: Incorrect calculation.
[C] :x: Incorrect calculation.
[D] ✓: Correct calculation of power.
## LEARNING AIDS
	**Common Mistakes**:
		1. Forgetting to integrate acceleration to find velocity: Always remember that velocity is the integral of acceleration with respect to time.
		2. Incorrectly performing the dot product: Ensure you multiply corresponding components and sum them up.
	**Quick Tips**:
		1.  Remember the formula for power as the dot product of force and velocity: $P = \vec{F} \cdot \vec{v}$.
		2. When integrating, remember the constant of integration, in this case it was assumed zero.

VENKATESWARARAO · Answer

$\left( {{\rm{2}}{{\rm{t}}^2}\,\,{\rm{ + }}\,{\rm{3}}{{\rm{t}}^3}\,} \right){\rm{W}}$

VENKATESWARARAO · Answer

$\left( {{\rm{2}}{{\rm{t}}^2}\,\,{\rm{ + }}\,4{{\rm{t}}^4}\,} \right){\rm{W}}$

VENKATESWARARAO · Answer

$\left( {{\rm{2}}{{\rm{t}}^3}\,\,{\rm{ + }}\,3{{\rm{t}}^4}\,} \right){\rm{W}}$

VENKATESWARARAO · Answer

The correct Option is "$\left( {{\rm{2}}{{\rm{t}}^3}\,\,{\rm{ + }}\,3{{\rm{t}}^5}\,} \right){\rm{W}}$"

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