The Question is from neet-physics laws-of-motion force, 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

NEET Physics Laws of Motion Force MCQs

Select Topic

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

Question Tags

NEET Subjects

Our Bots

Company