If the velocity v of a body is expressed as v = at + bt 2 , wh... | General Science MCQ | Examoogle

General SciencePhysics: Mechanics

If the velocity v of a body is expressed as v = at + bt 2, where a and b are constants, the dimensions of b are ____________.

$[LT^{-1}]$

$[L]$

$[LT^{-3}]$

$[LT^{-2}]$

Correct Answer

🔬 SC • Science Concept & PrincipleGeneral SciencePhysics: Mechanics

Option C

$[LT^{-3}]$

Quick Summary:

According to the Principle of Homogeneity of dimensions, the dimensions of each term in an equation must be the same. Since the given equation is $v = at + bt^2$ , the dimension of the term $bt^2$ must be equal to the dimension of velocity $v$ .

🔬SCScience SolutionConcept & Principle

💡 Explanation

💡 Everyday Analogy (Real-World Intuition)

Think of it like adding fruits; you can add apples to apples, but you cannot add apples to meters because they represent different 'physical' types. Dimensionally, everything on both sides of an equals sign must represent the same physical quantity.

🧠 Memory Mnemonic / Shortcut Aid

HOMO: Homogeneity Of Measurements means Only like terms add.

🔢 Key Formulas

$[v] = [LT^{-1}]$ — Dimensions of velocity

$[bt^2] = [LT^{-1}]$ — Principle of homogeneity applied to the term $bt^2$

$[b] = [LT^{-3}]$ — Derived dimensions of the constant b

⚙️ Working Principle

The principle states that we can only add or subtract physical quantities that have the same dimensions. Given that velocity $v$ has the dimension $[LT^{-1}]$ , the individual components $at$ and $bt^2$ must also have the dimension $[LT^{-1}]$ . Therefore, by setting $[bt^2] = [LT^{-1}]$ , we can isolate $[b]$ as $[b] = [LT^{-1}] / [T^2] = [LT^{-3}]$ .

📌 Key Points

▸
The Principle of Homogeneity is a fundamental requirement for the consistency of physical equations.
▸
Physical quantities must possess identical dimensions to be added or subtracted.
▸
Dimension analysis is a powerful tool to verify the correctness of physical formulas.

✅ Advantages

▸
Helps check the dimensional consistency of an equation.
▸
Allows derivation of relations between physical quantities.

❌ Disadvantages / Limitations

▸
Cannot determine dimensionless constants.
▸
Does not reveal the presence of scalar or vector nature.

🛠️ Applications / Uses

▸
Checking validity of physics equations.
▸
Converting physical quantities from one unit system to another.

📄 Additional Information

▸
Constant $a$ would have dimensions $[LT^{-2}]$ , representing acceleration.
▸
Option A $[LT^{-1}]$ is the dimension of velocity.
▸
Option B $[L]$ is the dimension of length.
▸
Option D $[LT^{-2}]$ is the dimension of acceleration.

📊 Diagram / Illustration

✅

C is correct — Based on the Principle of Homogeneity, the dimension of $[bt^2]$ must equal the dimension of velocity $[LT^{-1}]$ , which simplifies to $[LT^{-3}]$ .

Core Concepts Used

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Dimensional Analysis Principle of Homogeneity Kinematics

💡 EXAM TIP

Always ensure the power of T is calculated correctly in division: dividing by $T^2$ means subtracting the exponent, so $-1 - 2 = -3$ .

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