Stopping Distance And Time
The Math Is Simple
By: James R. Davis
After reading the TIP entitled You Only Hit The Car If You Don't Quite Stop In Time a person sent me a gentle critique of it as follows:
The general points you make are fine but I think you might want to check your math..
Traffic Engineers have some rules-of-thumb they developed over time. They, for example, have found that if the street surface is dry, the average person can safely decelerate an automobile at the rate of 15 feet per second (ft/sec/sec). That is, an average person can slow down at this rate without any real likelihood that they will loose control in the process.
The measure of velocity is distance divided by time (fps). The measure of acceleration (or deceleration in this case) is feet/sec/sec in the units you chose.
I believe he was trying to be helpful and was not just taking shots.
As to the measure of deceleration being fpsps rather than fps, I take no issue with that. My article said that you could '... decelerate .. at the rate of 15 fps' , but I think it is clear from the context that what I was saying was that regardless of the velocity, say it starts at 88 fps, you could scrub off 15 fps every second. i.e., after 1 second your velocity would be 73 fps, after 2 seconds it would be 58 fps, etc. For the mathematically inclined it would have been more accurate to say 'ft/sec/sec' rather than 'fps', but possibly more confusing to some.
[Engineers usually use a deceleration rate of 11.2 ft/sec/sec as noted in "A Policy on Geometric Design of Highways and Streets", Pg. 111. This is ultra conservative as the following quote from pg. 111 of that book demonstrates:
quote:You will note that it says that 'most drivers decelerate at a rate greater than 14.8 ft/sec/sec' and that more than 90% of drivers decelerate at a rate greater than 11.2 ft/sec/sec.]
He went on to say:
It would mean that you could stop your motorcycle in a total of 5.4 seconds (including the 1 second delay.) and your total stopping distance would be only 281.5 feet!
If you'll look at any road test of a current production motorcycle you'll see that stopping distances from 60 mph are typically 120 - 140 feet. Cages are frequently in the 150 - 180 foot range.
As to his suggestion that I recheck my math, I did, and obtained the same results.
So that there is no confusion, my message argued the point that by increasing your braking skills you could significantly reduce both the time it takes to stop and the distance taken to stop your motorcycle. Further, though I acknowledged that a motorcycle racer could get 1g deceleration (32 fpsps), or more, a reasonably skilled rider could easily get deceleration rates in excess of 20 fpsps. And, by contrast, showed what Traffic Engineers use as an assumption of safely attainable deceleration rates by the average person (15 fpsps).
So, I was not saying that you should (or can) try to get 1g deceleration rates, but that you can and should get much better braking (safely) than 'average' with just a little practice.
As to the numbers...
To determine how long it will take you to stop assuming a constant rate of deceleration, you need only divide your starting velocity (in fps) by your rate of deceleration.
60 MPH = 88 fps. (fps=1.467 * MPH). If your deceleration rate is 20 fpsps, then stopping time = 88/20 = 4.4 seconds. Since there is a 1 second delay in hitting your brakes (recognition and reaction time), the total time to stop is 5.4 seconds, just as I said.
To determine how far you will travel while braking you take 1/2 the starting velocity and multiply the result by the stopping time (ie, you calculate your average speed and multiply by how long you are moving.) In the cited case, this works out to be:
.5 * 88 * 4.4 = 193.6 feet. Since we traveled 88 feet before we hit the brakes, we add that to 193.6 and end up with a total of 281.6 feet, as I said (missed by .1 feet.)
So, how can my numbers be so far off from those reported? Simple. Clearly they are reporting JUST stopping distance and with deceleration rates of about 1 g. [Rider magazine once reported the results of stopping a Yamaha from 60 MPH AVERAGED 87 feet in a series of nine attempts. That was stopping at the rate of over 44 fps/s or approximately 1.3 g.]
Assuming a deceleration rate of 32 fpsps (1g), we calculate a braking stop time of 2.75 seconds (88/32). Distance traveled now is calculated to be 121 feet. (Ignoring the additional 88 feet you traveled before applying your brakes.) This is consistent with published reports, as he presented them.
The math is easy, the message is too - Skillful braking can save your life.
For those of you that are into math, I full well realize that I used an approximation for distance traveled when I simplified my formula and assumed an 'average speed'. Since the correct formula which would take the deceleration rate into account is beyond some of the readers, I chose to make it simple - because the message is also simple. (Besides, it yields the same answer.)
If you are interested, to calculate the distance using deceleration rates you would use:
x = x0 + (v0 * t) - (1/2 * a * t²)
x = distance traveled (feet)
x0 = starting distance (feet - for example, recognition/reaction
distance of 88)
v0 = initial velocity (feet per second)
t = stopping time (seconds)
a = deceleration rate (feet per second per second)
Copyright © 1992 - 2013 by The Master Strategy Group, all rights reserved.
(James R. Davis is a recognized expert witness in the fields of Motorcycle Safety/Dynamics.)