Units & Dimensions

A recipe says “add 2 flour.” Two what? Tablespoons? Cups? Bags? The number alone tells you nothing. You need the unit — the thing that says “what kind” and “how much of one.”

Physics is the same. Every measurement is a number plus a unit. Drop the unit and the number means nothing. NASA learned this the hard way in 1999 when the Mars Climate Orbiter burned up in the Martian atmosphere because one team used pounds and another used newtons. A $125 million spacecraft, lost to a unit mix-up.

The MCAT won’t ask you about NASA, but it will test whether you can:

• Convert between units in a few seconds.
• Spot answer choices that have the wrong dimensions.
• Use unit-checking to recover a forgotten formula or catch a math slip.

This is one of those sections where small habits pay off on every other physics problem you’ll ever do.

SI Base Units

The International System of Units (SI) defines seven base units. For the MCAT, you need six:

Derived Units

Most units you’ll see on the MCAT are built by combining the base units. You don’t need to memorize them all from scratch — recognize the common ones and remember how they’re built:

Metric Prefixes

The MCAT loves throwing prefixes at you — sometimes three different ones in the same passage. You need these in muscle memory:

Dimensional Analysis

Dimensional analysis is the single most powerful error-checking tool in physics. The idea: in any valid equation, the units on the left side must match the units on the right side. If they don’t, the equation is wrong. Period.

The method, in three steps:

1. Write the quantity you’re starting with, including its units.
2. Multiply by conversion factors arranged so the unwanted units cancel.
3. Check that only the desired units remain.

Example 1 — speed conversion. Convert 72 km/h to m/s.

$72 \,\frac{\text{km}}{\text{h}} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{72{,}000}{3600} \text{ m/s} = 20 \text{ m/s}$

The “km” cancels in step 2, the “h” cancels in step 3, and you’re left with m/s.

Example 2 — using units to catch a wrong answer. A passage asks for the kinetic energy of a 2 kg ball moving at 3 m/s. You scramble and write $KE = mv$, getting $2 \times 3 = 6$ kg·m/s. Pause and check the units: kg·m/s is momentum, not energy. Energy needs an extra factor of velocity. The correct formula is $KE = \tfrac{1}{2}mv^2$, giving $\tfrac{1}{2}(2)(9) = 9$ J. Dimensional analysis just saved you a wrong answer.

Using Dimensional Analysis to Recover a Formula

If you blank on a formula mid-test, dimensional analysis can often rebuild it.

Suppose you forget the formula for kinetic energy but remember:

• Energy is in joules: kg·m²/s².
• The formula must involve mass (kg) and velocity (m/s).

The only way to combine kg and m/s to get kg·m²/s² is mass times velocity squared: $mv^2$. The factor of $\tfrac{1}{2}$ doesn’t come from units (it comes from calculus), so on the MCAT you might still be off by a small constant — but you’ll be in the right ballpark and you’ll pick the right answer choice.

This trick rescues people on test day more often than they admit.

Common MCAT Unit Conversions

Keep these in your mental toolkit — they show up every test: