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Question

Which of the following equations is dimensionally homogeneous?

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: [A]
## CONCEPT FRAMEWORK
	**Primary Concept**: Dimensional Analysis
	**Related Topics**: Physical Quantities, Units and Measurements
	**Prerequisites**: Understanding of fundamental dimensions (Length [L], Time [T], Mass [M]) and how they relate to physical quantities.
## SOLUTION PATHWAY
**Given Information**:
	* We need to identify the dimensionally homogeneous equation from the given options.
	* Dimensional homogeneity means that all terms in an equation must have the same dimensions.
**Method & Steps**:
	1.  Identify the dimensions of each physical quantity:
		* v (velocity) = [LT⁻¹]
		* u (initial velocity) = [LT⁻¹]
		* a (acceleration) = [LT⁻²]
		* t (time) = [T]
		* s (displacement) = [L]
	2.  Analyze each option by substituting dimensions:
		* **Option A:** v = u + at
		   Dimensions: [LT⁻¹] = [LT⁻¹] + [LT⁻²][T] => [LT⁻¹] = [LT⁻¹] + [LT⁻¹].  This is dimensionally homogeneous.
		* **Option B:** s = ut + at
		   Dimensions: [L] = [LT⁻¹][T] + [LT⁻²][T] => [L] = [L] + [LT⁻¹]. This is NOT dimensionally homogeneous.
		* **Option C:** v² = u² + a
		   Dimensions: [LT⁻¹]² = [LT⁻¹]² + [LT⁻²] => [L²T⁻²] = [L²T⁻²] + [LT⁻²]. This is NOT dimensionally homogeneous.
		* **Option D:** s = vt²
		   Dimensions: [L] = [LT⁻¹][T²] => [L] = [LT]. This is NOT dimensionally homogeneous.
	3.  Conclude that only option A is dimensionally homogeneous.
		Result: Option A is the only dimensionally homogeneous equation.
 ## OPTION ANALYSIS
[A] ✓: The dimensions on both sides of the equation v = u + at are [LT⁻¹], making it dimensionally homogeneous.
[B] :x: The dimensions on the right side of the equation s = ut + at are a mixture of [L] and [LT⁻¹] making it not dimensionally homogeneous.
[C] :x: The dimensions on the right side of the equation v² = u² + a are a mixture of [L²T⁻²] and [LT⁻²] making it not dimensionally homogeneous.
[D] :x: The dimensions on the right side of the equation s = vt² are [LT] not [L] making it not dimensionally homogeneous.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Forgetting to square the dimensions when dealing with squared terms like v² or u².
		[How to avoid]: Always apply the power to the dimensions as well.
		2.[Error 2]: Ignoring the units of constants.
		[How to avoid]: In dimensional analysis, the numerical values are ignored. Focus only on the fundamental dimensions.
	**Quick Tips**:
		1.[Helpful shortcut]:  Check if all terms in an equation have the same dimensions. If not, the equation is not dimensionally homogeneous.
		2.[Memory aid]: Remember the fundamental dimensions: Length [L], Time [T], Mass [M]. Express all other quantities in terms of these dimensions.

NAGA SAI · Answer

The correct Option is "v = u + at"

NAGA SAI · Answer

s = ut + at

NAGA SAI · Answer

v^2 = u^2 + a

NAGA SAI · Answer

s = vt^2

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