What Is Energy? — Question 5

Question 5

An energy claim must survive a dimensional audit

A sensor reports mass $m$, speed $v$, acceleration $a$ and duration $t$. A proposed energy readout is $E=c\,m^\alpha v^\beta$, where c is dimensionless. Use $[E]=\mathrm{kg\,m^2/s^2}$; do not assume a numerical value for c.

1. Determine $\alpha$ and $\beta$.

2. Can dimensional analysis determine c?

3. Classify the dimensions of $ma$ and $mav$, and explain why neither is an energy.

Question 5 - Solution

An energy claim must survive a dimensional audit

(a) Match independent dimensions.

The governing relation is $$\boxed{[E]=M L^2T^{-2}=[m^\alpha v^\beta]=M^\alpha L^\beta T^{-\beta}}$$

$$[m^\alpha v^\beta]=\mathrm{kg}^{\alpha}\mathrm m^{\beta}\mathrm s^{-\beta},$$ so $\alpha=1$, $\beta=2$. The candidate is proportional to $mv^2$.

(b) Keep dimensionless physics separate.

No. Every dimensionless c gives the same units. A dynamical derivation or measurement must supply the coefficient; dimensional agreement alone does not establish a physical law.

(c) Count time factors.

$ma$ has force units $\mathrm{kg\,m/s^2}$. $mav$ has units $$\mathrm{kg\,m^2/s^3}=\mathrm{J/s},$$ a rate of energy transfer. Multiplying a rate by a suitable elapsed time can produce an energy.