The Question is from neet-physics laws-of-motion , and it was created by the author ABDUL JILANI BASHA

Question

A particle moves along a path defined by the equation $r = 2t^2\hat{i} + (3t - 1)\hat{j}$, where $r$ is in meters and $t$ is in seconds.  At $t = 2$ seconds, a force $F = (4t\hat{i} + 6\hat{j})$ Newtons acts on the particle.  What is the instantaneous power delivered by the force at $t = 2$ seconds?

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Instantaneous Power
	**Related Topics**: Velocity, Force, Dot Product
	**Prerequisites**: Vector Calculus, Newton's Laws
## SOLUTION PATHWAY
**Given Information**:
	* Position vector: $r = 2t^2\hat{i} + (3t - 1)\hat{j}$
	* Force vector: $F = (4t\hat{i} + 6\hat{j})$
	* Time: $t = 2$ seconds
	* Units are in meters, seconds, and Newtons.
**Method & Steps**:
	1. Find the velocity vector by taking the derivative of the position vector with respect to time.
		Reasoning: Velocity is the rate of change of position.
		Formula/Rule: $v = \frac{dr}{dt}$
	2. Calculate the velocity vector at t=2 seconds.
		Working: $v = \frac{d}{dt}(2t^2\hat{i} + (3t - 1)\hat{j}) = 4t\hat{i} + 3\hat{j}$. At $t=2$, $v = 4(2)\hat{i} + 3\hat{j} = 8\hat{i} + 3\hat{j}$
		Key Point: Remember to differentiate each component.
	3. Calculate the force vector at t=2 seconds.
		Working: $F = 4(2)\hat{i} + 6\hat{j} = 8\hat{i} + 6\hat{j}$
		Key Point: Substitute the given time into the force equation.
	4. Calculate the instantaneous power delivered by the force using the dot product of force and velocity vectors.
		Reasoning: Instantaneous power is the dot product of the force and velocity vectors.
		Formula/Rule: $P = F \cdot v$
	5. Calculate the dot product.
		Working: $P = (8\hat{i} + 6\hat{j}) \cdot (8\hat{i} + 3\hat{j}) = (8)(8) + (6)(3) = 64 + 18 = 82$
		Result: The instantaneous power is 82 Watts.
		Verification: The units are consistent (N*m/s = Watts).
 ## OPTION ANALYSIS
[A] :x: 64 Watts is the result of only considering the i components.
[B] ✓: 82 Watts is the correct calculation of the dot product.
[C] :x: 100 Watts is not the result of any common mistake.
[D] :x: 48 Watts is not the result of any common mistake.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. [Error 1]: Forgetting to use the dot product, instead just multiplying the magnitudes. [How to avoid]: Remember power is a scalar quantity, so use dot product for vector multiplication.
		2. [Error 2]: Not differentiating the position vector to get the velocity vector. [How to avoid]: Remember that velocity is the derivative of position with respect to time.
	**Quick Tips**:
		1. [Helpful shortcut]: Dot product of two vectors $A= a_1\hat{i} + a_2\hat{j}$ and $B = b_1\hat{i} + b_2\hat{j}$ is given by $A\cdot B = a_1b_1 + a_2b_2$.
		2. [Memory aid]: Power is the rate of doing work. Instantaneous power is $P = F \cdot v$

ABDUL JILANI BASHA · Answer

64 Watts

ABDUL JILANI BASHA · Answer

The correct Option is "82 Watts"

ABDUL JILANI BASHA · Answer

100 Watts

ABDUL JILANI BASHA · Answer

48 Watts

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