Why Does Height Formula Use -16 Instead of -32?

Date: 01/31/2008 at 17:52:42
From: Andy
Subject: why does the falling object model use 16 (not 32)

h = -16t^2 + s is given as the position of a falling object model, but
32ft per sec^2 is the acceleration of gravity.  If something is 
falling with the acceleration of gravity (32ft per sec^2) why to find
the position do you use 16 ft per sec^2?

I haven't had physics (I'm an alg II student), and I can't connect 
this to real life.  Can you explain it?

Date: 01/31/2008 at 19:57:23
From: Doctor Rick
Subject: Re: why does the falling object model use 16 (not 32)

Hi, Andy.

The coefficient of the t^2 term is HALF the acceleration.  That's the
reason for the 16.

When you get to calculus, you will learn how this works (the 
acceleration is the second derivative of at^2, which is 2a).

Before you get there, you can get the idea by constructing a table. 
Suppose an object starts at rest and begins falling at t=0.  The 
instantaneous speed after one second is 32 feet per second; the 
*average* speed over the first second (from t=0 to t=1) is (0+32)/2 
= 16 feet/sec.  If the average speed over each 1-second interval 
increases by 32 feet per second from one second to the next, then we 
can tabulate the distance moved in each second, and add them up to 
get the position at the end of each 1-second interval:

  Time interval   Avg. speed   Dist moved   Cumulative distance
  -------------   ----------   ----------   -------------------
  0 to 1 second    16 ft/sec      16 feet       16 feet
  1 to 2 seconds   48 ft/sec      48 feet       64 feet
  2 to 3 seconds   80 ft/sec      80 feet      144 feet
  3 to 4 seconds  112 ft/sec     112 feet      256 feet

What formula gives the position as a function of the elapsed time? 
It's 16 times the square of the number of seconds:

  after 1 second:  16 * 1^2 =  16 feet
  after 2 seconds: 16 * 2^2 =  64 feet
  after 3 seconds: 16 * 3^2 = 144 feet
  after 4 seconds: 16 * 4^2 = 256 feet

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

Date: 02/01/2008 at 14:32:33
From: Andy
Subject: Thank you (why does the falling object model use 16 (not 32))

Thank you for the explanation, it was exactly what I needed.  I still
have some thinking to do about the 16 for the first second but 32 for
every subsequent second, but it is starting to make more sense.  Can't
wait to get to calculus!

Date: 02/01/2008 at 15:26:04
From: Doctor Rick
Subject: Re: Thank you (why does the falling object model use 16 (not 32))

Hi, Andy.

I'll say a bit more about the 16 for the first second.  You know that 
constant acceleration means the velocity is increasing at a constant 
rate.  Suppose we broke it down into 1/10 second intervals.  If the 
velocity is 0 at t=0, and 32 at t=1, then it will be as follows at 
the 1/10 second intervals:

   t     v
  ---------
  0.0   0.0
  0.1   3.2
  0.2   6.4
  0.3   9.6
  0.4  12.8
  0.5  16.0
  0.6  19.2
  0.7  22.4
  0.8  25.6
  0.9  28.8
  1.0  32.0

You could average all these values the long way--or you could notice 
that the first plus the last equals 32.0, and the second plus the 
next-to-last also equals 32.0, and so on.  If we average the two 
values in any of those pairs, we'll get 32.0/2 = 16.0.  Then we can 
average these averages; they are all the same, so the average must 
be 16.0.

In general, for uniformly accelerated motion, the average velocity 
over an interval is equal to the average of the initial and final 
velocities.  And in general (not just for uniformly accelerated 
motion), the distance traveled is equal to the average velocity 
multiplied by the time interval.  That's how I built my table.

Calculus will extend this thinking by breaking down the interval 
further and further, approaching infinitely many parts.  Such 
thinking will then become exact, and will be put on solid ground. 
But for constant acceleration, it turns out that the correct result 
is exactly the same as what I have shown you by some slightly fuzzy 
arguments.

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

Date: 02/01/2008 at 15:51:30
From: Andy
Subject: Thank you (why does the falling object model use 16 (not 32))

Fuzzily getting clearer (a lot) - thanks for getting down to my level.