# Realistic Freeways

Freeways is an extremely cool game, in which the player designs interchanges for one-lane roadways. This page is an index for some more-or-less-realistic renditions of solved Freeways puzzles, organized by puzzle ID number and by minimum speed.

## Assumptions

• Slow (red), intermediate (green), and fast (blue) roadways support speeds of 36 ft/s (approximately 25 mi/h or 40 km/h), 72 ft/s, and 108 ft/s, respectively. Yellow and cyan roadways accommodate acceleration and deceleration.

• Side friction factor is 9 ft/s ÷ speed, superelevation is 1/12, and gravitational acceleration is 32 ft/s2.

• Deceleration is 12 ft/s2 and acceleration is 6 ft/s2.

• The Greenshields traffic model is used: speed = free-flow speed × (1 − jam density ÷ density). Jam density is 1 vehicle per 18 feet (same as vehicle length). The effects of merging and diverging are ignored.

• Each roadway, along with its traffic, occupies 12 ft of horizontal space and 24 ft of vertical space. Maximum grade is 1/48. Curves cannot be combined with acceleration or deceleration, but they can be combined with grades, as long as the grade does not begin or end on the curve.

## Notes

• Obviously, these simulations should not be used in real-life design. (Gap acceptance? Offtracking? Shoulders? Never heard of 'em.)

• Depicting acceleration and deceleration in SVG animation would be rather complicated (each individual car would have to be a separate element), so each yellow or cyan roadway instead uses its space mean speed to inform its CSS animation.

• Nominal scale is one pixel per foot, but the image won't become blurry if you zoom in or out: this is SVG, not GIF. (Elevated roadways are narrower than they should be, in order to avoid blocking the view of what's beneath them.) Precision is to the nearest one-tenth of a foot (i. e., ±0.05 ft).

• Maximum flow is achieved when density is half of jam density (i. e., 1 vehicle per 36 feet). The bottleneck of an interchange is maximum flow at minimum speed (108 ft/s ÷ 36 ft/veh = 3 veh/s, 72 ft/s ÷ 36 ft/veh = 2 veh/s, or 36 ft/s ÷ 36 ft/veh = 1 veh/s), which may occur at the start of an acceleration roadway, at the end of a deceleration roadway, or at the intersection of multiple roadways: not segments, but points, must be evaluated.

• Centripetal acceleration a = v2/rg(f + e) → rv2/g/(f + e) = v2 ÷ 32 ft/s2 ÷ (9 ft/s ÷ v + 1/12) = v3/(288 ft2/s3 + 8/3 ft/s2 × v). The minimum (centerline) radius of a slow roadway is 121.5 ft, that of an intermediate roadway is 777.6 ft, and that of a fast roadway is 2187 ft. (All curves are circular, not spiral.)

• If a symmetrical reverse curve joining two parallel line segments has radius r and produces an offset of y, each of that curve's two circular arcs has a length (not along the arc, but perpendicular to the line segments) of r√(1 − (1 − y/(2r))2). For example, if a car traveling at 108 ft/s wants to switch between two adjacent 12-ft lanes, each of the two arcs in the maneuver takes 2187 ft × √(1 − (1 − 12 ft ÷ (2 × 2187 ft))2) ≈ 161.9 ft of longitudinal space.

• The space required for acceleration or deceleration is (v12v02)/(2a). For example, if a car wants to decelerate from 108 ft/s to 72 ft/s, the maneuver requires ((72 ft/s)2 − (108 ft/s)2)/(2 × −12 ft/s2) = 270 ft.

## Puzzle 1

### Slow

#### T

Bounding box: 3937.1 ft × 583.5 ft ≈ 2 300 000 ft2