There's a formula that applies to everything. The nuance is in how coarse or fine that formula matches the real world.

The easy equation is really the area of a triangle:

Area = 1/2*length*height. That's because the area under a speed-time graph = the distance travelled; and if acceleration starts at zero and goes in a straight line, then this is a triangle.

But if it's not really a triangle, but is a series of different triangles, like "slow off the line, but hotter in second gear..." then you'd break up the triangles into increments; like time to 30, time from 30 to 60, time from 60 to 100. That gives you a more refined distance calculation over a series of small straight lines that can be calculated by arithmetic. Car magazines sometimes report these time intervals in performance comparisons.

In the real world, these small segments of straight lines are defined by a curve, you integrate the curve over time and come up with the distance. This is the difference between a coarse approximation by arithmetic, a finer approximation via calculus, and a measured value (within tolerances) achieved by repeated measurements over time and averaged out; or statistics.

To answer your basic question, the acceleration CAN be approximated by an average, it simply depends on your assumptions:

slow off the line: shift your acceleration (max speed/time value (ft/s^2)) down a bit to account for going slower longer [shallow increase early]

Steady pull the whole way: acceleration will be closer to speed/time [a straight line]

Fast off the line: shift acceleration up [steep increase early]

What do you guys do for a living that allows you to remember this much math and physics?

I took this stuff in high school and college, but that was about 25 years ago!

I remember nothing.

I couldn't solve this problem if you gave me $20 million dollars.

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Paul S.

2001 E430, Bourdeaux Red, Oyster interior.
79,200 miles.

1973 280SE 4.5, 170,000 miles. 568 Signal Red, Black MB Tex. "The Red Baron".

The tragedy with all scientific endeavors is that the questions limit the answers.

We only solve for what we seek.

Forcing bovines into a parallelepiped is one excellent example.

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Here's a pic. The curves more/less have to be like this based on the two fixed data points: 0-100 and 20 seconds (within reason, to simplify the concept...).

The distance travelled is blue. The green lines are notional times taken to get to a certain speed. You could use these to be more accurate.

The "average" acceleration is the slope of the red line: shallow if you're slow off the line, steep if you're fast off the line [call it 4 mph/sec for slow -- 100/25; and 6 mph/sec for fast -- 100/16; 5 mph/sec for average]. It's an average slope since there's some blue above the line but also some white below the line.

This "average" is the basis for most responses' calculations, since that's really all they've got to go on.

Distance = 1/2*average acceleration*time^squared

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Professional Engineers & Post Secondary Teachers (in maths /science / engineering).
A sound understanding of equations of motion & calculus are important in most profession engineering.
From the posts on this thread it is clear what level of maths various members studied up to. Problem is when some one dismisses something that they dont understand because of lack of knowledge of maths.

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Grumpy Old Diesel Owners Club group

I no longer question authority, I annoy authority. More effect, less effort....

1967 230-6 auto parts car. rust bucket.
1980 300D now parts car 800k miles
1984 300D 500k miles
1987 250td 160k miles English import
2001 jeep turbo diesel 130k miles
1998 jeep tdi ~ followed me home. Needs a turbo.
1968 Ford F750 truck. 6-354 diesel conversion.
Other toys ~J.D.,Cat & GM ~ mainly earth moving

And here I always thought math clubs would be boring

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1982 300GD Carmine Red (DB3535) Cabriolet Parting Out
1990 300SEL Smoke Silver (Parting out)
1991 350SDL Blackberry Metallic (481)

"The thing is Bob, its not that I'm lazy...its that I just don't care."

I just thought I would approach this problem in the direction it was going, i.e. downhill

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Strelnik
Invest in America: Buy a Congressman!

1950 170SD
1951 Citroen 11BN
1953 Citroen 11BNF limo
1953 220a project
1959 180D
1960 190D
1960 Borgward Isabella TS 2dr
1983 240D daily driver
1983 380SL
1990 350SDL daily driver alt
3 x Citroen DS21M, down from 5
3 x Citroen 2CV, down from 6

Uh, ditto. But I agree that the only way the "correct" answer as given in this thread can be assumed "correct" is if rate of acceleration were constant. I knew that much (that the info required to answer the question accurately really isn't there), but I had no idea what that formula was. Too long ago!

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1992 300D 2.5T
1980 Euro 300D (sadly, sold)
1998 Jetta TDI, 132K "Rudy"
1974 Triumph TR6
1999 Saab 9-5 wagon (wife's)

Thank you all for your input. Based on that, can I conclude that to get the most accurate result, an actual graph of the acceleration of the vehicle used in the example would have to be used?

It would also seem that using the average rate of acceleration will also provide a fairly acceptable answer.

I'm usually pretty good at algebra and geometry but Calculus is the first and last class I ever failed in school...

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-Evan


Benz Fleet:
1968 UNIMOG 404.114
1998 E300
2008 E63


Non-Benz Fleet:
1992 Aerostar
1993 MR2
2000 F250

The answer is: How much accuracy do you need?

If you need 1 min accuracy, do the math.

If you need 1 sec accuracy, get the timer and run the drag racing lights.