# Confirmation of a formula

1. Dec 5, 2012

### NewDarkAge

Hi

I'm doing a poster on stopping distances for Physics, but it's a topic we haven't really discussed yet in the lessons. In my research I found this formula:

"d = V^2/(2g(f + G))

Where:
d = Braking Distance
g = Acceleration due to gravity
G = Roadway grade as a percentage;
V = Initial vehicle speed
f = Coefficient of friction between the tires and the roadway"

I just wanted to check with you guys whether this is a valid formula as I haven't found it anywhere else. Also, I was wondering if you could give me any ideas for what to include in the poster; I'm really looking for formulae relating different factors that affect braking distance.

I hope this all makes sense, and thanks very much for your time

S

2. Dec 5, 2012

### CWatters

The equation looks about right.

If you start with one of the standard equations of motion and ignore the grade you can get to this equation reasonably easily..

d = V2 /2gμ

where
d = stopping distance
V = Initial velocity
g = acc due to gravity
μ = coefficient of friction

Along the way you use:

force = mass * acceleration
and
friction force = Normal force * coefficient of friction
which is equivalent to
friction force = Mass * g * coefficient of friction

Note how the definition of coefficient of friction relates vertical to horizontal forces. So (without doing the maths) it's not unreasonable that u term might be modified by the grade which is also expressed as a ratio of vertical to horizontal.

Perhaps another person fancies doing the full proof for you.

3. Dec 5, 2012

### NemoReally

You need to be careful to use the correct units in these equations and the range of validity (eg, G is derived from the small angle approximation of the sine of the grade). Have a look at the following document for a bit more detail ... https://engineering.purdue.edu/~flm/CE%20361_files/chapter1_2_notes_.pdf [Broken] (around about Problem 2.21).

A couple of items to think about (depending upon what the intent of the poster is) are reaction time (If it takes you 1.5 seconds to react how far will the car travel before you even hit the brakes!) and wheel lockup (you want to skedaddle away from having an accident not skidaddle into one)

Last edited by a moderator: May 6, 2017
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