JEE PYQ: Units & Measurements - Question ID 8b42665875c5 (JEE Main 2020)

In dimensional analysis, we express any physical quantity in terms of fundamental quantities (mass, length, time, etc.). Here, the problem redefines the fundamental quantities as momentum ($P$), area ($A$), and time ($T$). Our goal is to express energy in terms of these new fundamental dimensions.

The key formulas we use are:

• Momentum: $P = mv$ (mass × velocity)
• Area: $A = L^2$ (length squared)
• Energy (kinetic): $E = \frac{1}{2}mv^2$

We need to express mass ($M$), length ($L$), and velocity ($v$) in terms of $P$, $A$, and $T$ to rewrite energy.

Step 1: Express mass ($M$) in terms of $P$, $A$, and $T$

From momentum: $P = mv \implies m = \frac{P}{v}$. But we need $m$ in terms of $P$, $A$, and $T$ only. So, we need to express $v$ in terms of $A$ and $T$.

Step 2: Express velocity ($v$) in terms of $A$ and $T$

Velocity is distance over time: $v = \frac{L}{T}$. But $A = L^2 \implies L = A^{1/2}$. Thus, $v = \frac{A^{1/2}}{T}$.

Step 3: Substitute $v$ back into the expression for mass

$m = \frac{P}{v} = \frac{P}{A^{1/2}/T} = P A^{-1/2} T$.

Step 4: Express energy ($E$) in terms of $P$, $A$, and $T$

Kinetic energy: $E = \frac{1}{2}mv^2$. Substitute $m = P A^{-1/2} T$ and $v = A^{1/2} T^{-1}$:

Simplify the expression:

The constant $\frac{1}{2}$ is dimensionless and can be ignored in dimensional analysis. Thus, the dimensional formula for energy is:

Step 5: Match with the given options

The derived formula matches option C: $\left[ P A^{1/2} T^{-1} \right]$.

Students often make the following mistakes:

• Incorrect substitution of velocity: Forgetting that $v = \frac{L}{T}$ and $L = A^{1/2}$ leads to wrong expressions for mass and energy.
• Ignoring dimensional consistency: Some students directly equate energy to $P^2$ or $A$ without proper derivation, leading to incorrect options like A or B.
• Sign errors in exponents: Misapplying the rules of exponents (e.g., $(A^{1/2})^2 = A^1$, not $A$) can result in wrong dimensional formulas.

Exam Tip: Always express all intermediate quantities (like mass and velocity) in terms of the given fundamental quantities before substituting into the target formula. Double-check exponent arithmetic to avoid sign errors.