A physicist uses dimensional analysis to derive an expression for the energy (E) of a particle based on its mass (m), velocity (v), and Planck's constant (h). They arrive at $E = mv^2$. Why is this result incomplete?

Which of the following cannot be determined using dimensional analysis?

The vernier scale of a vernier callipers has $10$ divisions that coincide with $9$ divisions on the main scale. If the main scale is marked in millimeters, what is the least count of the instrument (in cm)?

In a vernier callipers, $(N+3)$ divisions of vernier scale coincide with $N$ divisions of the main scale. If 1 MSD represents $0.1$ mm, the vernier constant (in cm) is:

In a vernier callipers, $21$ divisions on the vernier scale coincide with $20$ divisions on the main scale. If each division on the main scale is $0.5$ mm, find the vernier constant (in cm).

Dimensional analysis can help in deriving relationships between physical quantities. However, it cannot:

Dimensional analysis can be used to check the correctness of an equation. However, it fails to identify the presence of:

Dimensional analysis is used to check the validity of an equation relating the speed of sound (v) in a gas to its pressure (P) and density ($\rho$). Which limitation prevents dimensional analysis from distinguishing between $v = \sqrt{P/\rho}$ and $v = \sqrt{14P/\rho}$?

Dimensional analysis fails to determine the formula for a physical quantity if it depends on:

A student uses dimensional analysis to analyze the time ($au$) it takes for a capacitor (C) to discharge through a resistor (R). They correctly identify that $au$ depends on R and C. However, they are unable to derive the precise relationship $au = RC$. What limitation of dimensional analysis explains this?

Which of these equations, though dimensionally correct, might be physically incorrect due to limitations of dimensional analysis?