Units and Dimensional Checks

If there is one absolute truth about the PE Civil Transportation exam, it is this: Unit errors will sink your score.

Transportation engineering is heavily reliant on empirical formulas (formulas derived from observation rather than pure physics). Because of this, equations frequently mix units—taking a speed in miles per hour (mph) and a time in seconds (s) to produce a distance in feet (ft).

Mastering unit conversions and using dimensional checks is the easiest way to prevent unforced errors.

The “Target Unit First” Method

Before you write down a single number or look up a single formula, read the very last sentence of the problem statement and write down the target unit.

Example Problem: “What is the required stopping sight distance for a vehicle traveling at 55 mph on a 3% downgrade, given a perception-reaction time of 2.5 seconds?”

Identify the Target: The problem asks for “distance.” Unless specified otherwise, sight distance in US Customary units is measured in feet (ft).
Write it Down: On your scratchpad, write [Target: ft] and draw a box around it.
Align the Givens: Now look at your givens: speed is in mph, time is in s. You instantly know a conversion factor will be required to get from miles and hours to feet and seconds.

The Most Critical Transportation Conversions

You should have these conversions memorized so thoroughly that you don’t even need to think about them:

Speed: $1 \text{ mph} = 1.47 \text{ ft/sec}$
(Technically it is $1.4667...$, but $1.47$ is standard across AASHTO and the PE exam. When you see $1.47$ in a formula, know that it is simply a unit converter turning $V$ from mph into fps).
Distance: $1 \text{ mile} = 5,280 \text{ feet}$
Time: $1 \text{ hour} = 3,600 \text{ seconds}$
Area: $1 \text{ acre} = 43,560 \text{ square feet}$
Volume: $1 \text{ cubic yard} (CY) = 27 \text{ cubic feet} (CF)$

The Trap of Empirical Formulas

In pure physics, if you have an equation like $F = ma$, the units on the left must perfectly balance the units on the right.

In Transportation, empirical formulas do not follow this rule. They are built assuming you input specific units.

Example: AASHTO Minimum Radius Formula

$$ R = \frac{V^2}{15(0.01e + f)} $$

Where:

$R$ = radius in feet
$V$ = speed in mph
$e$ = superelevation in percent
$f$ = side friction factor

If you plug $V$ in as feet per second (fps), the equation will output garbage. The constant $15$ in the denominator actually contains the built-in conversion factors to resolve the mismatch between mph and feet.

How to survive empirical formulas: Whenever you find a formula in the HCM, AASHTO, or the Reference Handbook, read the variable definitions below it. Do not assume $V$ is always mph. Sometimes the standard defines $v$ (lowercase) as fps and $V$ (uppercase) as mph.

Dimensional Analysis (The Built-In Checker)

If you are using a standard mathematical formula (not an empirical one), carrying your units through the calculation is a foolproof way to catch algebraic mistakes.

Example: Flow Rate You are given a density ($k$) of $40 \text{ vehicles/mile}$ and a speed ($u$) of $50 \text{ mph}$. What is the flow ($q$) in $\text{vehicles/hour}$?

Formula: $q = u \times k$
Plug in with units: $q = \left( 50 \frac{\text{miles}}{\text{hour}} \right) \times \left( 40 \frac{\text{vehicles}}{\text{mile}} \right)$
Cancel the units: The “miles” in the numerator and denominator cancel out.
Result: $2000 \frac{\text{vehicles}}{\text{hour}}$.

Because the resulting unit ($\text{veh/hr}$) matches your target unit, you can be highly confident your algebra is correct. If your dimensional check resulted in $\text{vehicles} \cdot \text{hours} / \text{miles}^2$, you immediately know you multiplied instead of dividing somewhere in your setup.

Practice Habit

For the next 10 practice problems you solve, do not write down a single number on your scratch paper without writing its unit right next to it. It takes an extra 0.5 seconds per line and will save you from failing by a single point.