Gravity Model

Transportation · FE Reference Handbook section

Transportation

6 formulas

10 exam-style examples

~57 min

All Transportation lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Gravity Model within Transportation. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what gravity model describes physically and when it applies.
State every one of the 6 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: grades as decimals in curve formulas, percent in the stem.

Lecture

Why this section exists. Gravity Model is the part of Transportation that lets you connect a vertical or horizontal alignment, or a traffic stream to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a curve geometry element or a capacity/flow relationship. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. grades as decimals in curve formulas, percent in the stem. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Dense peak-hour traffic queued on an urban arterial at dusk. — Photo 1. Where this shows up in practice: gravity model.
Wikimedia Commons, CC BY 2.0

Transportation — Gravity Model: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a vertical or horizontal alignment, or a traffic stream. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 6 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

Tij	Quantity produced by "Tij = Pi > /" — read its definition and unit from the handbook line directly above the equation.
Pi	Quantity produced by "Pi = total number of trips produced in Zone i" — read its definition and unit from the handbook line directly above the equation.
Aj	Quantity produced by "Aj = number of trips attracted to Zone j" — read its definition and unit from the handbook line directly above the equation.
Fij	Quantity produced by "Fij = friction factor that is an inverse function of travel time between Zones i and j" — read its definition and unit from the handbook line directly above the equation.
Kij	Quantity produced by "Kij = socioeconomic adjustment factor for travel between Zones i and j" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

A jFijKij
A jFijKij H
where

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Trip distribution between two zones with the gravity model — Gravity Model

Zone i produces 3,099 trips. Zone 1 has 2,421 attractions with a friction factor of 0.80; zone 2 has 1,327 attractions with a friction factor of 0.45. Use the gravity model to distribute the trips.

Given

P_i = 3,099 trips
A₁ = 2,421, F₁ = 0.80
A₂ = 1,327, F₂ = 0.45

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2421 (0.80) = 1, 937$
Numerator (zone 2)
$1327 (0.45) = 597.2$
Denominator
$1, 937 + 597.2 = 2, 534$
Substituting — T_i1 = 3099(1,937/2,534) = 2,369 trips
Balance — T_i2 = 3099 − 2,369 = 730.3 trips

Answer: Zone 1 receives 2,369 trips; zone 2 receives 730.3 trips

Why the other options are there

1,550 trips each (attractions ignored)
2,002 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 2

Trip distribution between two zones with the gravity model — Gravity Model (2)

Zone i produces 3,353 trips. Zone 1 has 2,976 attractions with a friction factor of 0.70; zone 2 has 2,476 attractions with a friction factor of 0.70. Use the gravity model to distribute the trips.

Given

P_i = 3,353 trips
A₁ = 2,976, F₁ = 0.70
A₂ = 2,476, F₂ = 0.70

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2976 (0.70) = 2, 083$
Numerator (zone 2)
$2476 (0.70) = 1, 733$
Denominator
$2, 083 + 1, 733 = 3, 816$
Substituting — T_i1 = 3353(2,083/3,816) = 1,830 trips
Balance — T_i2 = 3353 − 1,830 = 1,523 trips

Answer: Zone 1 receives 1,830 trips; zone 2 receives 1,523 trips

Why the other options are there

1,677 trips each (attractions ignored)
1,830 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 3

Trip distribution between two zones with the gravity model — Gravity Model (3)

Zone i produces 2,510 trips. Zone 1 has 2,203 attractions with a friction factor of 0.40; zone 2 has 2,910 attractions with a friction factor of 0.45. Use the gravity model to distribute the trips.

Given

P_i = 2,510 trips
A₁ = 2,203, F₁ = 0.40
A₂ = 2,910, F₂ = 0.45

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2203 (0.40) = 881.2$
Numerator (zone 2)
$2910 (0.45) = 1, 310$
Denominator
$881.2 + 1, 310 = 2, 191$
Substituting — T_i1 = 2510(881.2/2,191) = 1,010 trips
Balance — T_i2 = 2510 − 1,010 = 1,500 trips

Answer: Zone 1 receives 1,010 trips; zone 2 receives 1,500 trips

Why the other options are there

1,255 trips each (attractions ignored)
1,081 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 4

Trip distribution between two zones with the gravity model — Gravity Model (4)

Zone i produces 3,520 trips. Zone 1 has 1,974 attractions with a friction factor of 0.35; zone 2 has 886 attractions with a friction factor of 0.95. Use the gravity model to distribute the trips.

Given

P_i = 3,520 trips
A₁ = 1,974, F₁ = 0.35
A₂ = 886, F₂ = 0.95

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1974 (0.35) = 690.9$
Numerator (zone 2)
$886 (0.95) = 841.7$
Denominator
$690.9 + 841.7 = 1, 533$
Substituting — T_i1 = 3520(690.9/1,533) = 1,587 trips
Balance — T_i2 = 3520 − 1,587 = 1,933 trips

Answer: Zone 1 receives 1,587 trips; zone 2 receives 1,933 trips

Why the other options are there

1,760 trips each (attractions ignored)
2,430 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 5

Trip distribution between two zones with the gravity model — Gravity Model (5)

Zone i produces 3,940 trips. Zone 1 has 2,276 attractions with a friction factor of 0.65; zone 2 has 817 attractions with a friction factor of 0.45. Use the gravity model to distribute the trips.

Given

P_i = 3,940 trips
A₁ = 2,276, F₁ = 0.65
A₂ = 817, F₂ = 0.45

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2276 (0.65) = 1, 479$
Numerator (zone 2)
$817 (0.45) = 367.7$
Denominator
$1, 479 + 367.7 = 1, 847$
Substituting — T_i1 = 3940(1,479/1,847) = 3,156 trips
Balance — T_i2 = 3940 − 3,156 = 784.2 trips

Answer: Zone 1 receives 3,156 trips; zone 2 receives 784.2 trips

Why the other options are there

1,970 trips each (attractions ignored)
2,899 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 6

Trip distribution between two zones with the gravity model — Gravity Model (6)

Zone i produces 2,033 trips. Zone 1 has 1,914 attractions with a friction factor of 0.50; zone 2 has 1,264 attractions with a friction factor of 0.85. Use the gravity model to distribute the trips.

Given

P_i = 2,033 trips
A₁ = 1,914, F₁ = 0.50
A₂ = 1,264, F₂ = 0.85

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1914 (0.50) = 957.0$
Numerator (zone 2)
$1264 (0.85) = 1, 074$
Denominator
$957.0 + 1, 074 = 2, 031$
Substituting — T_i1 = 2033(957.0/2,031) = 957.8 trips
Balance — T_i2 = 2033 − 957.8 = 1,075 trips

Answer: Zone 1 receives 957.8 trips; zone 2 receives 1,075 trips

Why the other options are there

1,017 trips each (attractions ignored)
1,224 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 7

Trip distribution between two zones with the gravity model — Gravity Model (7)

Zone i produces 3,459 trips. Zone 1 has 1,703 attractions with a friction factor of 0.85; zone 2 has 2,165 attractions with a friction factor of 0.90. Use the gravity model to distribute the trips.

Given

P_i = 3,459 trips
A₁ = 1,703, F₁ = 0.85
A₂ = 2,165, F₂ = 0.90

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$1703 (0.85) = 1, 448$
Numerator (zone 2)
$2165 (0.90) = 1, 949$
Denominator
$1, 448 + 1, 949 = 3, 396$
Substituting — T_i1 = 3459(1,448/3,396) = 1,474 trips
Balance — T_i2 = 3459 − 1,474 = 1,985 trips

Answer: Zone 1 receives 1,474 trips; zone 2 receives 1,985 trips

Why the other options are there

1,730 trips each (attractions ignored)
1,523 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 8

Trip distribution between two zones with the gravity model — Gravity Model (8)

Zone i produces 3,781 trips. Zone 1 has 713 attractions with a friction factor of 0.75; zone 2 has 1,915 attractions with a friction factor of 0.95. Use the gravity model to distribute the trips.

Given

P_i = 3,781 trips
A₁ = 713, F₁ = 0.75
A₂ = 1,915, F₂ = 0.95

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$713 (0.75) = 534.8$
Numerator (zone 2)
$1915 (0.95) = 1, 819$
Denominator
$534.8 + 1, 819 = 2, 354$
Substituting — T_i1 = 3781(534.8/2,354) = 858.9 trips
Balance — T_i2 = 3781 − 858.9 = 2,922 trips

Answer: Zone 1 receives 858.9 trips; zone 2 receives 2,922 trips

Why the other options are there

1,891 trips each (attractions ignored)
1,026 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 9

Trip distribution between two zones with the gravity model — Gravity Model (9)

Zone i produces 1,557 trips. Zone 1 has 2,868 attractions with a friction factor of 0.70; zone 2 has 2,225 attractions with a friction factor of 0.90. Use the gravity model to distribute the trips.

Given

P_i = 1,557 trips
A₁ = 2,868, F₁ = 0.70
A₂ = 2,225, F₂ = 0.90

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2868 (0.70) = 2, 008$
Numerator (zone 2)
$2225 (0.90) = 2, 003$
Denominator
$2, 008 + 2, 003 = 4, 010$
Substituting — T_i1 = 1557(2,008/4,010) = 779.5 trips
Balance — T_i2 = 1557 − 779.5 = 777.5 trips

Answer: Zone 1 receives 779.5 trips; zone 2 receives 777.5 trips

Why the other options are there

778.5 trips each (attractions ignored)
876.8 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Example 10

Trip distribution between two zones with the gravity model — Gravity Model (10)

Zone i produces 2,230 trips. Zone 1 has 2,226 attractions with a friction factor of 0.70; zone 2 has 2,621 attractions with a friction factor of 0.80. Use the gravity model to distribute the trips.

Given

P_i = 2,230 trips
A₁ = 2,226, F₁ = 0.70
A₂ = 2,621, F₂ = 0.80

Find

Trips distributed to each zone

Start with the thinking

The gravity model shares productions in proportion to attraction × friction factor.
The denominator always sums every competing destination, so the shares total 100%.

Step-by-step solution

Formula — T_ij = P_i · (A_j F_ij) / Σ(A_k F_ik)
Numerator (zone 1)
$2226 (0.70) = 1, 558$
Numerator (zone 2)
$2621 (0.80) = 2, 097$
Denominator
$1, 558 + 2, 097 = 3, 655$
Substituting — T_i1 = 2230(1,558/3,655) = 950.7 trips
Balance — T_i2 = 2230 − 950.7 = 1,279 trips

Answer: Zone 1 receives 950.7 trips; zone 2 receives 1,279 trips

Why the other options are there

1,115 trips each (attractions ignored)
1,024 trips (friction factors ignored)

Reference: FE Reference Handbook — Transportation → Gravity Model

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a vertical or horizontal alignment, or a traffic stream, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Gravity Model contains 6 relations; you must be able to find this page in under 15 seconds.
Exam style: a curve geometry element or a capacity/flow relationship.
Unit rule: grades as decimals in curve formulas, percent in the stem.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

grades as decimals in curve formulas, percent in the stem
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.

Gravity Model

Learning objectives

Lecture

Theory, developed

Notation used in this section

Handbook notes for this section

Core formulas for this FE topic

Tij=Pi>/

Tij=numberoftripsthatareproducedinZoneiandattractedtoZonej

Pi=totalnumberoftripsproducedinZonei

Aj=numberoftripsattractedtoZonej

Fij=frictionfactorthatisaninversefunctionoftraveltimebetweenZonesiandj

Kij=socioeconomicadjustmentfactorfortravelbetweenZonesiandj

Worked exam-style examples

Self-check

Chapter summary

Common traps in this section