Tonight we discuss the
acceleration due to gravity - technically, "local gravity". It has a symbol (g), and it is
approximately equal to 9.8 m/s/s, near the surface of the Earth. At higher
altitudes, it becomes lower - a related phenomenon is that the air pressure
becomes less (since the air molecules are less tightly constrained), and it
becomes harder to breathe at higher altitudes (unless you're used to it). Also,
the boiling point of water becomes lower - if you've ever read the "high
altitude" directions for cooking Mac n Cheese, you might remember that you
have to cook the noodles longer (since the temperature of the boiling water is
lower).
On the Moon, which is
a smaller body (1/4 Earth radius, 1/81 Earth mass), the acceleration at the
Moon's surface is roughly 1/6 of a g (or around 1.7 m/s/s). On Jupiter, which
is substantially bigger than Earth, the acceleration due to gravity is around
2.2 times that of Earth. All of these things can be calculated without ever
having to visit those bodies - isn't that neat?
Consider the meaning
of g = 9.8 m/s/s. After 1 second of freefall, a ball would achieve a speed of
.....
9.8 m/s
After 2 seconds....
19.6 m/s
After 3 seconds....
29.4 m/s
We can calculate the
speed by rearranging the acceleration equation:
vf = vi + at
In this case, vf is
the speed at some time, a is 9.8 m/s/s, and t is the time in question. Note
that the initial velocity is 0 m/s. In fact, when initial velocity is 0, the expression is really simple:
vf = g t
Got it?
The distance is a bit
trickier to figure. This formula is useful - it comes from combining the
definitions of average speed and acceleration.
d = vi t + 0.5 at^2
Since the initial
velocity is 0, this formula becomes a bit easier:
d = 0.5 at^2
Or....
d = 0.5 gt^2
Or.....
d = 4.9 t^2
(if you're near the
surface of the Earth, where g = 9.8 m/s/s)
This is close enough
to 10 to approximate, so:
d = 5 t^2
So, after 1 second, a
freely falling body has fallen:
d = 5 m
After 2 seconds....
d = 20 m
After 3 seconds....
d = 45 m
After 4 seconds...
d = 80 m
This relationship is
worth exploring. Look at the numbers for successive seconds of freefall:
0 m
5 m
20 m
45 m
80 m
125 m
180 m
If an object is
accelerating down an inclined plane, the distances will follow a similar
pattern - they will still be proportional to the time squared. Galileo noticed
this. Being a musician, he placed bells at specific distances on an inclined
plane - a ball would hit the bells. If the bells were equally spaced, he (and
you) would hear successively quickly "dings" by the bells. However,
if the bells were located at distances that were progressively greater (as
predicted by the above equation, wherein the distance is proportional to the
time squared), one would hear equally spaced 'dings."
Check this out:
Equally spaced bells:
http://www.youtube.com/watch?v=06hdPR1lfKg&feature=related
Bells spaced
according to the distance formula:
http://www.youtube.com/watch?v=totpfvtbzi0
Furthermore, look at
the numbers again:
0 m
5 m
20 m
45 m
80 m
125 m
180 m
Each number is
divisible by 5:
0
1
4
9
16
25
36
All perfect squares,
which Galileo noticed - this holds true on an inclined plane as well, and its
easier to see with the naked eye (and time with a "water clock.")
Look at the
differences between successive numbers:
1
3
5
7
9
All odd numbers.
Neat, eh?
FYI:
http://www.mcm.edu/academic/galileo/ars/arshtml/mathofmotion1.html
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