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# The Mathematics Behind Stopping a Car

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Question: if it takes 20 feet to stop a car going 20 MPH, how far does it take to stop a car going 40 MPH?
1. 10 feet.
2. 20 feet.
3. 40 feet.
4. 80 feet.

The answer, which surprises nearly everyone, is (4) 80 feet (neglecting the driver's reaction time). This is because the energy of a moving car is proportional to its mass times the square of its velocity, or:

$\displaystyle e = m v^2$
Where:
• e = energy
• m = mass
• v = velocity

A practical embodiment of this equation that takes into account a typical driver's reaction time, is:

$\displaystyle d = 1.5 \frac{5280}{3600} v + \frac{v^2}{20}$
Where:
• d = total stopping distance, feet
• v = vehicle speed, miles per hour

A concise version of the above equation, with the absolute minimum number of terms, is:

$\displaystyle d = 2.2 v + \frac{v^2}{20}$

This equation measures real-world distances, thus it uses field measurements, which will differ to some extent based on their source. It predicts stopping distance in feet for a given velocity in miles per hour. A reaction time of 1.5 seconds is allowed for the driver to commence stopping (Hey! I didn't invent the car radio!). The factor 5280/3600 converts the distanced traveled while reacting into feet per second. Here is a table of typical values, which were generated using this equation and which agree closely with data published by public safety organizations:

 Speed MPH Reaction Distance Feet Vehicle Distance Feet Total Distance Feet 20 44 20 64 30 66 45 111 40 88 80 168 50 110 125 235 60 132 180 312 70 154 245 399 80 176 320 496 90 198 405 603 100 220 500 720

Virtually no one realizes that a car's stopping distance increases as the square of velocity. Ordinarily, not knowing physics and math is inconvenient. But in this case it can get you killed.

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