Could This Average Speed Get You Fined? The Calculus Says Yes

Last Updated on June 18, 2026 by Statnzee Team

The Big Idea: Does Traveling 170 Miles in 2 Hours Prove I Drove 85 mph?

Suppose a driver is accused of speeding because:

• Total distance traveled = 170 miles
• Total time taken = 2 hours

Someone calculates:

Average Speed = Distance ÷ Time
              = 170 ÷ 2
              = 85 mph

The question is:

Does this prove that the driver was driving at exactly 85 mph at some point during the trip?

The Mathematical Answer

Yes — provided the vehicle’s speed changed continuously and there were no impossible jumps in speed.

This conclusion comes from a result in calculus called the Mean Value Theorem (MVT).

Understanding the Mean Value Theorem

Let:

s(t) = distance traveled at time t

At the start:

s(0) = 0 miles

After 2 hours:

s(2) = 170 miles

Average speed over the trip is:

(s(2) - s(0)) / (2 - 0)
= 170 / 2
= 85 mph

The Mean Value Theorem says that if the distance function is smooth and continuous, then there must exist some moment c between 0 and 2 such that:

s'(c) = 85 mph

Since:

s'(t) = instantaneous speed

there must have been at least one instant when the vehicle’s speed was exactly 85 mph.

Why This Does Not Mean the Driver Was Always Driving 85 mph

The driver may have driven:

• 60 mph for some time
• 90 mph for some time
• 80 mph for some time
• Stopped at a traffic signal

Yet the theorem guarantees that somewhere during the journey the speedometer must have shown exactly 85 mph.

A Simple Example

Suppose the driver travels:

• First hour at 70 mph → 70 miles
• Second hour at 100 mph → 100 miles

Total:

70 + 100 = 170 miles

Average speed:

170 ÷ 2 = 85 mph

Since speed increased from 70 mph to 100 mph, the speedometer must have passed through:

71, 72, 73, ...
84, 85, 86, ...
99, 100

Therefore there was certainly a moment when the car was traveling exactly 85 mph.

Can This Be Used as Proof for a Speeding Ticket?

Not necessarily.

The Mean Value Theorem proves only that:

At some instant the speed was 85 mph.

It does not tell us:

• When that happened
• Where it happened
• Whether the speed limit there was lower than 85 mph
• How long the driver maintained that speed

In real life, traffic authorities usually rely on radar, cameras, GPS records, or speed sensors rather than average-speed calculations alone.

Business Application #1: Delivery Companies

Suppose a logistics company records:

• Delivery route length = 340 miles
• Completion time = 4 hours

Average speed:

340 ÷ 4 = 85 mph

Managers can conclude that at some point the truck was traveling at 85 mph.

This helps identify:

• Potential speeding behavior
• Safety risks
• Fuel-efficiency issues

Business Application #2: E-Commerce Delivery Optimization

Companies like courier services monitor delivery performance.

If a vehicle covers a long route unusually quickly, analytics systems can estimate average speeds.

The Mean Value Theorem implies that certain speeds must have been reached during the trip, helping auditors investigate route compliance.

Business Application #3: Manufacturing

Suppose a factory produces:

850 units in 10 hours

Average production rate:

850 ÷ 10 = 85 units/hour

The Mean Value Theorem says there was at least one moment when the production line was operating at exactly:

85 units/hour

Managers use this idea to study machine performance and efficiency.

Business Application #4: Website Traffic Analytics

Imagine a website receives:

8,500 visitors in 100 hours

Average traffic rate:

85 visitors/hour

Then there must have been some instant when traffic was arriving at exactly 85 visitors per hour.

Digital marketers use this principle when analyzing:

• Advertising campaigns
• Server capacity planning
• User engagement trends

Key Takeaway

The Mean Value Theorem tells us:

If a quantity changes continuously, then somewhere along the way its instantaneous rate of change must equal its average rate of change.

For the driving example:

170 miles in 2 hours
Average speed = 85 mph

Therefore:

There was at least one instant during the trip when the vehicle’s speed was exactly 85 mph.

This beautiful calculus result connects mathematics to transportation, logistics, manufacturing, finance, and business analytics.