Acceleration Formula and Squared Units

Date: 01/24/2005 at 11:11:27
From: Missy
Subject: acceleration formula and answer in square units

In the acceleration formula a = (f-i)/t, a = acceleration, f = final 
speed, i = initial speed, and t = time.  

"The speed of a car changes from 37ft/s to 102 ft/s in five 
seconds.  What is its acceleration in feet/second squared?"

Why is the answer for acceleration written in square units?

Date: 01/24/2005 at 11:30:36
From: Doctor Edwin
Subject: Re: acceleration formula and answer in square units

Hi, Missy.

This is one of my pet peeves.  It's very easy to get confused about 
this, and (at least in my school) nobody did a good job explaining it.

I'm going to try a different approach: Suppose I had a train that was 
accelerating very, very slowly.  Right now it's going 1 foot/second 
and it is gaining speed at 1ft/s every hour.  So in one hour it will 
be going 2ft/s, in two hours it will be going 3ft/s, etc.

So it is increasing speed at one ft/s every hour.  That is, 1 foot per 
second per hour or 1ft/sec/hr. If you write that out, you get

    1 foot     1
    ------ *  ----
     sec      1 hr

or

     1 foot
    ---------
     sec * hr

That's the same as 1/60th of a ft/sec/min:

     1 foot
    -------------
    60 sec * min

or 1/3600 of a ft/sec/sec

     1 foot
    --------------
    3600 sec * sec

or, to be extreme:

    .0002777 fs^-2

So at least until you're used to the notation, don't think of it as 
square units at all.  Think of it as speed per second--how much speed 
it loses or gains each second. 

Does that help at all?  Write back if you're still stuck!
     
- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 01/24/2005 at 11:56:07
From: Missy
Subject: acceleration formula and answer in square units

Dr. Edwin,

Thank you for the response and sample.  To clarify a bit further, 
are the square units only used in this instance because we are talking 
about time in seconds (and the number would be tedious to work with so
square units are substituted), or are the answers always written in
square units (regardless of the time component)?

Missy

Date: 01/24/2005 at 13:34:32
From: Doctor Edwin
Subject: Re: acceleration formula and answer in square units

Hi, Missy.

Well, to be honest I've never seen them written as anything other 
than square units, but I think that's just because I've never seen an 
acceleration that would be better expressed as feet/second per year 
(for example).  Maybe the movements of continents might be expressed 
in centimeters per year per century or something.  

But yes, in principle, I think it's true to say that we're only using
square time units because we're talking about speed per second and
we're talking about change per second at the same time, so we end up
with two "per seconds"'s in one phrase, and if you multiply it out you 
get "per second squared."

Did that help answer your question?  Write back if you want to talk 
more about this.

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/