14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace

Professional Traffic Operations Engineer PTOE

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1 Introduction & PTOE Exam Overview

1.1 The PTOE Credential: Certification Overview & Industry Value 1.2 Core Industry Reference Standards in Traffic Operations 1.3 Study Guide Architecture, Blueprint Alignment & Exam-Day Strategy

2 Domain 1: Traffic Flow Theory & Fundamental Concepts

2.1 Macroscopic Traffic Stream Models & Fundamental Diagram (q = k × u) 2.2 Microscopic Traffic Characteristics & Car-Following Models 2.3 Hydrodynamic Shockwave Analysis & Traffic Queuing Models

3 Domain 1: Highway & Interrupted Flow Capacity Analysis (HCM)

3.1 Basic Freeway Segments & Multilane Highways (HCM 7th Edition) 3.2 Freeway Weaving, Merging, and Diverging Segments 3.3 Signalized Intersection Capacity & Delay Modeling (HCM)

4 Domain 1: Managed Lanes, Transit Priority, TSMO & Travel Demand Management

4.1 Unsignalized Intersections (TWSC, AWSC) & Modern Roundabouts Capacity 4.2 Managed Lanes, HOV/HOT Operations, & Ramp Metering 4.3 Transit Priority & Multimodal Signal Operations (TSP) 4.4 Transportation Systems Management and Operations (TSMO) & Regional ITS 4.5 Travel Demand Management (TDM) & Trip Reduction Strategies

5 Domain 2: Geometric Design Controls, Criteria, & Sight Distance

5.1 Functional Classification, Design Controls, & Context-Sensitive Design 5.2 Sight Distance Principles: Stopping, Decision, & Passing (AASHTO) 5.3 Horizontal Alignment, Curve Radius, & Superelevation Design

6 Domain 2: Roadway Cross-Sections & Access Management

6.1 Vertical Alignment: Tangent Grades, Crest & Sag Vertical Curves 6.2 Roadway Cross-Section Elements & AASHTO Clear Zone Concepts 6.3 Access Management Principles & Driveway Spacing Criteria

7 Domain 2: Intersection & Interchange Geometric Design

7.1 Intersection Channelization, Turn Lane Design, & Storage Length 7.2 Interchange Configurations & Operational Comparisons (Diamond, Cloverleaf, SPUI) 7.3 Diverging Diamond Interchanges (DDI / DCMI) Operational Principles 7.4 Modern Roundabout Geometric Design Elements & Fastest Paths 7.5 Multimodal Geometric Design: Pedestrian Facilities, Bikeways, & ADA

8 Domain 3: Traffic Safety Concepts, Human Factors, & Safe System

8.1 The Safe System Approach & Vision Zero Principles 8.2 Human Factors, Driver Expectancy, & Perception-Reaction Time (PRT) 8.3 Crash Collision Typologies & Statistical Rate Formulations (MEV, MVMT)

9 Domain 3: Highway Safety Manual (HSM) Predictive Methods & EB

9.1 Highway Safety Manual (HSM) Predictive Methodology & Part C Models 9.2 Safety Performance Functions (SPFs) & Crash Modification Factors (CMFs) 9.3 Empirical Bayes (EB) Method & Regression-to-the-Mean (RTM) Correction 9.4 Economic Appraisal of Safety Projects (Benefit-Cost Analysis, NPV)

10 Domain 3: Safety Countermeasures & Road Safety Audits (RSA)

10.1 Roadway Departure Countermeasures (Rumble Strips, Safety Edge, Barriers) 10.2 Intersection Safety Countermeasures (Sight Triangles, Left-Turn Calming, Roundabouts) 10.3 Vulnerable Road User (VRU) Safety Countermeasures (PHB/HAWK, RRFB, Curb Extensions) 10.4 Road Safety Audits (RSAs) & Systemic Safety Analysis

11 Domain 4: Traffic Signal Warrants & Isolated Timing (MUTCD/ITE)

11.1 MUTCD Traffic Signal Warrants (Warrants 1 through 9) 11.2 Signal Phasing Principles & NEMA Dual-Ring Operations 11.3 Clearance Intervals (ITE Kinematic Equations) & Pedestrian Timing Parameters

12 Domain 4: Signal Coordination, Networks, & ATSPMs

12.1 Actuated Signal Controller Parameters (Passage Time, Gap Reduction, Max Recall) 12.2 Traffic Signal Coordination: Bandwidth, Offsets, & Progression Models 12.3 Traffic Responsive & Adaptive Traffic Signal Control Systems (ASCS) 12.4 Automated Traffic Signal Performance Measures (ATSPMs) & High-Resolution Data

13 Domain 4: Signing, Markings, Work Zones, Grade Crossings & School Zones (MUTCD)

13.1 MUTCD Regulatory, Warning, & Guide Signs Principles and Placement 13.2 Sign Retroreflectivity Standards & Nighttime Visual Maintenance 13.3 Longitudinal Pavement Markings, Delineators, & Object Markers 13.4 Temporary Traffic Control (TTC) & Work Zone Safety (MUTCD Part 6) 13.5 Highway-Rail Grade Crossings: Warning Devices, Preemption & Storage 13.6 School Zone Traffic Control & Student Walk Route Planning

14 Domain 5: Traffic Studies & Applied Statistical Methods

14.1 Applied Statistical Distributions & Sample Size Determination 14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace 14.3 Volume Count Methodologies & Expansion Factoring (ADT, AADT, PHF)

15 Domain 5: Traffic Impact Studies & Trip Generation (ITE)

15.1 Travel Time, Delay, & Intersection Queue Studies 15.2 Parking Supply, Duration, & Turnover Analysis 15.3 Traffic Impact Study (TIS) Scoping, Horizon Years, & Mitigation 15.4 ITE Trip Generation Handbook: Average Rates vs. Regression Equations

16 Domain 6: Environmental, Institutional, Ethics & Equity Issues

16.1 Vehicle Emissions, Air Quality Conformity, & Decarbonization 16.2 Traffic Noise Modeling (TNM) & Noise Abatement Criteria (NAC) 16.3 Long-Range Transportation Planning & the NEPA Environmental Review 16.4 Professional Engineering Ethics, Tort Liability, & Risk Management 16.5 Transportation Equity, Access & Multimodal Community Considerations

14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace

Key Takeaways Explain This Topic AI

Spot speed studies measure instantaneous vehicular speeds at a specific roadway location under free-flow conditions (headways >= 4 s) to evaluate operating speeds and support speed limit setting.

Radar and LiDAR measurements subject to angular placement experience cosine error, which underestimates true vehicle speed according to v true = v measured / cos ⁡ ( θ ) v_{\text{true}} = v_{\text{measured}} / \cos(\theta) v true ​ = v measured ​ / cos ( θ ) ; angles θ > 10 \theta > 10 θ > 10 degrees require explicit mathematical correction.

The 85th percentile speed (V_85) represents the speed at or below which 85% of drivers travel under free-flow conditions and serves as the primary engineering benchmark for setting statutory and regulatory speed limits per the MUTCD.

The 10-mph pace is the 10-mph speed range containing the greatest percentage of sampled vehicles, reflecting speed uniformity; when > 70% of vehicles fall within the pace, speed variance and collision risks are minimized.

Time Mean Speed ( v ˉ t \bar{v}_t v ˉ t ​ , arithmetic average of spot speeds) is always greater than or equal to Space Mean Speed ( v ˉ s \bar{v}_s v ˉ s ​ , harmonic mean based on travel time) per Wardrop's relationship: v ˉ t = v ˉ s + σ s 2 / v ˉ s \bar{v}_t = \bar{v}_s + \sigma_s^2 / \bar{v}_s v ˉ t ​ = v ˉ s ​ + σ s 2 ​ / v ˉ s ​ .

Last updated: August 2026

14.2 Spot Speed Studies, 85th Percentile Speed, & 10-mph Pace

PTOE Exam Focus: Spot speed studies are fundamental to Domain 5. Candidates must understand study design protocols, correct for radar/LiDAR cosine error ( v true = v measured cos ⁡ θ v_{\text{true}} = \frac{v_{\text{measured}}}{\cos\theta} v true ​ = c o s θ v measured ​ ​ ), construct and interpret cumulative frequency distribution curves, determine the 85th percentile speed ( V 85 V_{85} V 85 ​ ) for speed zone establishment, calculate the 10-mph pace and percentage in pace, and mathematically relate Time Mean Speed ( v ˉ t \bar{v}_t v ˉ t ​ ) to Space Mean Speed ( v ˉ s \bar{v}_s v ˉ s ​ ) via Wardrop's formulation.

1. Objectives & Field Protocols of Spot Speed Studies

A spot speed study records the instantaneous speeds of individual vehicles traversing a specific, localized roadway cross-section. Its primary engineering applications include:

Establishing and updating regulatory speed limits (speed zoning per MUTCD).

Assessing speed dispersion and crash risk correlation.

Evaluating the effectiveness of traffic calming countermeasures.

Sizing traffic signal change and clearance intervals.

Verifying design speed consistency and stopping sight distance adequacy.

Field Sampling Protocols:

Free-Flowing Conditions: Only sample vehicles with time headways ≥ 4 to 5 seconds \ge 4\text{ to }5\text{ seconds} ≥ 4 to 5 seconds . Do not sample following vehicles trapped in platoons.

Unbiased Selection: Sample every n n n -th vehicle or random free-flow vehicles across all lanes. Do not preferentially target high-speed outliers or sports cars.

Ideal Geometry & Environmental Conditions: Conduct studies on straight, level tangents away from driveways, signals, and intersections during dry, daylight off-peak hours.

Unobtrusive Observer Placement: Conceal the observer and equipment to prevent driver braking upon spotting surveillance.

2. Speed Measurement Technologies & Cosine Error Correction

Common spot speed instrumentation includes:

Radar / LiDAR Guns: Handheld Doppler radar ( 24.15 GHz 24.15\text{ GHz} 24.15 GHz K-band or 34.7 GHz 34.7\text{ GHz} 34.7 GHz Ka-band) and infrared LiDAR laser pulses ( 904 nm 904\text{ nm} 904 nm ). Handheld units must be aimed along the direction of travel.

Pneumatic Road Tubes: Dual rubber tubes placed across the lane separated by a fixed distance ( d = 10 to 16 ft d = 10\text{ to }16\text{ ft} d = 10 to 16 ft ). An electronic counter measures pulse time difference Δ t \Delta t Δ t , yielding v = d / Δ t v = d / \Delta t v = d /Δ t .

Video Analytics & Computer Vision: High-definition cameras calibrated with roadway ground control points tracking vehicle bounding boxes.

Cosine Error Mechanics & Correction

When a radar or LiDAR operator stands off the roadway shoulder, the line-of-sight vector forms an angle θ \theta θ with the vehicle's true trajectory. Because Doppler radar measures only the radial velocity component along the line of sight: v measured = v true ⋅ cos ⁡ θ v_{\text{measured}} = v_{\text{true}} \cdot \cos\theta v measured ​ = v true ​ ⋅ cos θ Roadway Centerline (True Vehicle Trajectory)

• ====================================[ Vehicle: v_true ]========================>
• ^ /
• \ / Line of Sight Vector
• \ / (Measured Velocity: v_measured)
• \ theta /
• \ /
• \ /
• [ Radar Gun Operator ] (Offset on Shoulder)

Critical Cosine Error Rule:

Cosine error always causes the device to underestimate true speed (since cos ⁡ θ ≤ 1.0 \cos\theta \le 1.0 cos θ ≤ 1.0 ). To recover the true vehicular velocity: v true = v measured cos ⁡ θ v_{\text{true}} = \frac{v_{\text{measured}}}{\cos\theta} v true ​ = cos θ v measured ​ ​

If θ ≤ 10 ∘ \theta \le 10^\circ θ ≤ 1 0 ∘ : cos ⁡ ( 10 ∘ ) = 0.9848 \cos(10^\circ) = 0.9848 cos ( 1 0 ∘ ) = 0.9848 , error is < 1.5 % < 1.5\% < 1.5% (acceptable without manual correction in routine screening).

If θ > 10 ∘ \theta > 10^\circ θ > 1 0 ∘ (e.g., θ = 25 ∘ \theta = 25^\circ θ = 2 5 ∘ to 30 ∘ 30^\circ 3 0 ∘ ): cos ⁡ ( 25 ∘ ) = 0.9063 \cos(25^\circ) = 0.9063 cos ( 2 5 ∘ ) = 0.9063 , error exceeds 9.4 % 9.4\% 9.4% and must be mathematically corrected .

3. Cumulative Speed Distribution & Percentile Metrics

Spot speed observations are grouped into frequency bins (typically 2 mph 2\text{ mph} 2 mph or 5 mph 5\text{ mph} 5 mph intervals) to generate a Cumulative Speed Distribution Curve (an S-shaped ogive curve plotted as Cumulative Percentage on the vertical axis versus Speed on the horizontal axis). Cumulative

• Percent (%)
• 100 ^ .---'
• | _.-'
• 85 |------------------------------------.-' | <-- 85th Percentile Speed (V_85)
• | .-' |
• 50 |-----------------------------.-' | <-- Median Speed (V_50)
• | .-' |
• 15 |-----------------------.-' | | <-- 15th Percentile Speed (V_15)
• | _.-' | |
• 0 +------------------+--------+------------+------------------------->
• 0 V_15 V_50 V_85 Speed (mph)
• |<-- 10-mph Pace ---->|

Critical Percentile Benchmarks:

85th Percentile Speed ( V 85 V_{85} V 85 ​ ): The speed at or below which 85 % 85\% 85% of free-flowing vehicles travel. It represents the speed that reasonable, prudent drivers adopt under prevailing roadway conditions. MUTCD Section 2B.35 mandates that posted regulatory speed limits be established within 5 mph 5\text{ mph} 5 mph of the 85th percentile speed.

50th Percentile Speed ( V 50 V_{50} V 50 ​ / Median): The speed that divides the distribution into two equal halves ( 50 % 50\% 50% drive slower, 50 % 50\% 50% drive faster).

15th Percentile Speed ( V 15 V_{15} V 15 ​ ): The speed at or below which only 15 % 15\% 15% of vehicles travel; used as an engineering guideline for establishing minimum speed limits or identifying slow-moving hazards.

• 4. The 10-mph Pace & Modal Speed
• Definition of the 10-mph Pace

The 10-mph Pace is the specific 10 mph 10\text{ mph} 10 mph speed increment (e.g., 42 to 52 mph 42\text{ to }52\text{ mph} 42 to 52 mph ) that contains the highest percentage of sampled vehicles across the entire distribution.

Percentage of Vehicles in the 10-mph Pace Percent in Pace = ( Number of Vehicles in 10-mph Pace Total Sample Size N ) × 100 % \text{Percent in Pace} = \left( \frac{\text{Number of Vehicles in 10-mph Pace}}{\text{Total Sample Size } N} \right) \times 100\% Percent in Pace = ( Total Sample Size N Number of Vehicles in 10-mph Pace ​ ) × 100%

Uniform Flow (Ideal): ≥ 70 % \ge 70\% ≥ 70% of vehicles in pace. Indicates high driver consensus and low speed variance.

Dispersed Flow (High Crash Risk): < 50 % to 60 % < 50\%\text{ to }60\% < 50% to 60% in pace. Indicates wide speed disparities, causing frequent overtaking, tailgating, and elevated collision rates.

Modal Speed

The Modal Speed is the single speed value occurring with the greatest frequency (the peak of the speed frequency histogram).

• 5. Speed Dispersion, Skewness, & Standard Deviation
• Standard Deviation ( s s s )

Quantifies the dispersion or spread of speeds around the sample mean v ˉ \bar{v} v ˉ : s = ∑ i = 1 N ( v i − v ˉ ) 2 N − 1 s = \sqrt{\frac{\sum_{i=1}^{N} (v_i - \bar{v})^2}{N - 1}} s = N − 1 ∑ i = 1 N ​ ( v i ​ − v ˉ ) 2 ​ ​

For an approximately normal distribution, standard deviation can be estimated from percentiles: s ≈ V 85 − V 15 2 ⋅ 1.036 ≈ V 85 − V 15 2.07 s \approx \frac{V_{85} - V_{15}}{2 \cdot 1.036} \approx \frac{V_{85} - V_{15}}{2.07} s ≈ 2 ⋅ 1.036 V 85 ​ − V 15 ​ ​ ≈ 2.07 V 85 ​ − V 15 ​ ​

Skewness Index

Evaluates the asymmetry of the speed distribution: Skewness = 2 ( V 85 − V 50 ) V 85 − V 15 − 1 \text{Skewness} = \frac{2(V_{85} - V_{50})}{V_{85} - V_{15}} - 1 Skewness = V 85 ​ − V 15 ​ 2 ( V 85 ​ − V 50 ​ ) ​ − 1

Zero Skewness ( 0.0 0.0 0.0 ): Perfectly symmetric normal distribution ( V 85 − V 50 = V 50 − V 15 V_{85} - V_{50} = V_{50} - V_{15} V 85 ​ − V 50 ​ = V 50 ​ − V 15 ​ ).

Positive (Right) Skew ( > 0 > 0 > 0 ): Mean and 85th percentile are pulled higher by a long tail of high-speed outliers.

Negative (Left) Skew ( < 0 < 0 < 0 ): Long tail of slow-moving vehicles.

6. Time Mean Speed ( v ˉ t \bar{v}_t v ˉ t ​ ) vs Space Mean Speed ( v ˉ s \bar{v}_s v ˉ s ​ )

Traffic engineering strictly distinguishes between point-based arithmetic speed and length-based harmonic speed:

1. Time Mean Speed ( v ˉ t \bar{v}_t v ˉ t ​ )

The arithmetic average of instantaneous spot speeds measured at a single point: v ˉ t = 1 N ∑ i = 1 N v i \bar{v}_t = \frac{1}{N} \sum_{i=1}^{N} v_i v ˉ t ​ = N 1 ​ i = 1 ∑ N ​ v i ​

2. Space Mean Speed ( v ˉ s \bar{v}_s v ˉ s ​ )

The harmonic average of speeds over a roadway segment length L L L , equivalent to total vehicle-distance divided by total vehicle-travel-time: v ˉ s = N ∑ i = 1 N 1 v i = L ⋅ N ∑ i = 1 N t i \bar{v}_s = \frac{N}{\sum_{i=1}^{N} \frac{1}{v_i}} = \frac{L \cdot N}{\sum_{i=1}^{N} t_i} v ˉ s ​ = ∑ i = 1 N ​ v i ​ 1 ​ N ​ = ∑ i = 1 N ​ t i ​ L ⋅ N ​

Wardrop's Equilibrium Relation: v ˉ t = v ˉ s + σ s 2 v ˉ s \bar{v}_t = \bar{v}_s + \frac{\sigma_s^2}{\bar{v}_s} v ˉ t ​ = v ˉ s ​ + v ˉ s ​ σ s 2 ​ ​

Where σ s 2 \sigma_s^2 σ s 2 ​ is the variance of the space mean speed distribution.

Fundamental Principle: Because variance is non-negative ( σ s 2 ≥ 0 \sigma_s^2 \ge 0 σ s 2 ​ ≥ 0 ), Time Mean Speed is always greater than or equal to Space Mean Speed ( v ˉ t ≥ v ˉ s \bar{v}_t \ge \bar{v}_s v ˉ t ​ ≥ v ˉ s ​ ). They are equal only when all vehicles travel at identically equal speeds ( σ s 2 = 0 \sigma_s^2 = 0 σ s 2 ​ = 0 ).

Spot Speed Metrics, Mathematical Formulations, and Engineering Applications

Speed Parameter Symbol / Formulation Typical Empirical Value Operational & Safety Significance

85th Percentile Speed V_85 (85% cumulative point) 45 - 65 mph (suburban arterial) Primary statutory basis for setting posted speed limits per MUTCD 2B.35

50th Percentile (Median) V_50 (50% cumulative point) 40 - 58 mph Measures central tendency; unaffected by extreme high/low outlier speeds

15th Percentile Speed V_15 (15% cumulative point) 34 - 50 mph Establishes lower boundary of reasonable operating speed / minimum speed limits

10-mph Pace 10-mph range with max vehicles e.g., 42 - 52 mph Identifies dominant operating speed cohort of the traffic stream

Percentage in 10-mph Pace (Vehicles in Pace / N) * 100% > 70% (good); < 55% (hazardous) Direct indicator of speed uniformity; low values correlate with high crash rates

Cosine Error Correction v true = v measured / cos ⁡ ( θ ) v_{\text{true}} = v_{\text{measured}} / \cos(\theta) v true ​ = v measured ​ / cos ( θ ) Correction factor: 1 / cos ⁡ ( θ ) 1 / \cos(\theta) 1/ cos ( θ ) Compensates for radar/LiDAR angle offset; uncorrected values underestimate true speed

Time Mean vs Space Mean v ˉ t = v ˉ s + σ s 2 / v ˉ s \bar{v}_t = \bar{v}_s + \sigma_s^2 / \bar{v}_s v ˉ t ​ = v ˉ s ​ + σ s 2 ​ / v ˉ s ​ v ˉ t \bar{v}_t v ˉ t ​ is 1 to 5 mph higher than v ˉ s \bar{v}_s v ˉ s ​ Fundamental traffic flow theory link between point speed and density (q = k * u_s)

Loading diagram... Spot Speed Distribution: Histogram, Cumulative Curve, and 10-mph Pace

Cumulative Vehicle Speed Distribution and Percentile Benchmarks (Sample Size N = 200)

• 7. Worked Calculation Examples
• Example 1: Radar Cosine Error Correction

Problem: A technician positions a handheld radar gun on the outer highway shoulder, creating an angle of θ = 25 ∘ \theta = 25^\circ θ = 2 5 ∘ between the radar line of sight and the center of the approaching travel lane. The radar unit displays a measured speed of v measured = 48.0 mph v_{\text{measured}} = 48.0\text{ mph} v measured ​ = 48.0 mph . What is the true operating speed of the vehicle?

Step-by-Step Solution:

Identify Formula: v true = v measured cos ⁡ θ v_{\text{true}} = \frac{v_{\text{measured}}}{\cos\theta} v true ​ = cos θ v measured ​ ​

Compute Trigonometric Cosine: cos ⁡ ( 25 ∘ ) = 0.9063 \cos(25^\circ) = 0.9063 cos ( 2 5 ∘ ) = 0.9063

Calculate True Speed: v true = 48.0 mph 0.9063 = 52.96 mph ≈ 53.0 mph v_{\text{true}} = \frac{48.0\text{ mph}}{0.9063} = 52.96\text{ mph} \approx 53.0\text{ mph} v true ​ = 0.9063 48.0 mph ​ = 52.96 mph ≈ 53.0 mph

Engineering Note: Failure to correct for the 25 ∘ 25^\circ 2 5 ∘ angle would result in underestimating the vehicle's speed by 5.0 mph 5.0\text{ mph} 5.0 mph .

Example 2: Time Mean Speed vs. Space Mean Speed Calculation

Problem: Four vehicles traverse a 1.0 -mile 1.0\text{-mile} 1.0 -mile roadway segment with measured spot speeds of 30 mph 30\text{ mph} 30 mph , 40 mph 40\text{ mph} 40 mph , 50 mph 50\text{ mph} 50 mph , and 60 mph 60\text{ mph} 60 mph . Compute the Time Mean Speed ( v ˉ t \bar{v}_t v ˉ t ​ ) and the Space Mean Speed ( v ˉ s \bar{v}_s v ˉ s ​ ).

Step-by-Step Solution:

Time Mean Speed ( v ˉ t \bar{v}_t v ˉ t ​ , Arithmetic Mean): v ˉ t = 30 + 40 + 50 + 60 4 = 180 4 = 45.0 mph \bar{v}_t = \frac{30 + 40 + 50 + 60}{4} = \frac{180}{4} = 45.0\text{ mph} v ˉ t ​ = 4 30 + 40 + 50 + 60 ​ = 4 180 ​ = 45.0 mph

Travel Times for Each Vehicle over L = 1.0 mile L = 1.0\text{ mile} L = 1.0 mile :

• t 1 = 1 / 30 = 0.03333 hr t_1 = 1 / 30 = 0.03333\text{ hr} t 1 ​ = 1/30 = 0.03333 hr
• t 2 = 1 / 40 = 0.02500 hr t_2 = 1 / 40 = 0.02500\text{ hr} t 2 ​ = 1/40 = 0.02500 hr
• t 3 = 1 / 50 = 0.02000 hr t_3 = 1 / 50 = 0.02000\text{ hr} t 3 ​ = 1/50 = 0.02000 hr
• t 4 = 1 / 60 = 0.01667 hr t_4 = 1 / 60 = 0.01667\text{ hr} t 4 ​ = 1/60 = 0.01667 hr

Total travel time ∑ t i = 0.03333 + 0.02500 + 0.02000 + 0.01667 = 0.09500 hr \sum t_i = 0.03333 + 0.02500 + 0.02000 + 0.01667 = 0.09500\text{ hr} ∑ t i ​ = 0.03333 + 0.02500 + 0.02000 + 0.01667 = 0.09500 hr

Space Mean Speed ( v ˉ s \bar{v}_s v ˉ s ​ , Harmonic Mean): v ˉ s = N ⋅ L ∑ t i = 4 × 1.0 0.09500 = 42.11 mph ≈ 42.1 mph \bar{v}_s = \frac{N \cdot L}{\sum t_i} = \frac{4 \times 1.0}{0.09500} = 42.11\text{ mph} \approx 42.1\text{ mph} v ˉ s ​ = ∑ t i ​ N ⋅ L ​ = 0.09500 4 × 1.0 ​ = 42.11 mph ≈ 42.1 mph

Verification: v ˉ t ( 45.0 ) > v ˉ s ( 42.1 ) \bar{v}_t (45.0) > \bar{v}_s (42.1) v ˉ t ​ ( 45.0 ) > v ˉ s ​ ( 42.1 ) , satisfying Wardrop's principle.