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| Perhaps the simplest form of projectile motion occurs
when we drop a ball and watch it bounce. But if we drop two balls,
together, one on top of the other, the result can be a little surprising,
particularly if the bottom ball is a basketball and the top ball is a baseball.
Some people call it baseketball.
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One of the possibly little known facts of mathematical
physics is that if you drop a ball from some height above the ground and shoot
another ball from ground level at the dropped ball, you will hit the dropped
ball if you aim directly at it (rather than be high with your shot). See
Shoot
the Monkey for an illustration of this. That this is perhaps a little
known fact of mathematical physics may be explained
by its inaccuracy. It fails to allow for the effect of air resistance
(drag) and how this alters speed and direction.
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| A nice animation involving the path of a golf ball
with air resistance taken as proportional to velocity can be found at Golf
Range. In this interactive animation the user can choose to include
the effect of air resistance or not. If you neglect air resistance than
the flight of the golf ball follows a symmetric parabolic path. It is
probably easier to predict where the ball will land in this case (and easier to
get a hole-in-one in the animation). If you consider the effect of air
resistance then the path is no longer symmetric about its midpoint. You
can visually observe how the ball is "knocked down" at the
end of its path. If air resistance is neglected |
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| then a golf
ball or baseball will travel the farthest for a given initial velocity
(basically, how hard it was hit) if it leaves the bat or golf club at a path
that makes a 45 degree angle with the horizontal. This is another fact of
mathematical physics that neglects air resistance. In fact if we consider
air resistance in the Golf
Range demonstration we find that the golf ball travels further if launched
at an initial angle of less than 45 degrees.
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Many factors come into play in determining how
far a batted baseball will travel. These factors include the density of
the air and condition of the baseball (which will effect air resistance), the
angle at which the ball is launched, the bat speed, and the speed of the
pitch. A wonderful animation illustrating the
path of a batted ball for different speeds of the pitch, bath speeds, and launch
angles can be found at Science
of Baseball. There islots of fun here for a baseball fan. Click
on Scientific Slugger
when you get to the site to play with the batted ball demonstration. |
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| If
you want to see whether or not your reaction time is quick enough to hit a 90
mph fastball try Fastball Reaction Time
on this same site. The Baseketball demonstration
above also comes from this site. |
Another example where air resistance plays a
major role is in jumping out of an airplane. The hope here is that the
parachute will open and there will be enough air resistance to limit the
speed of your descent. In fact mathematics can be used to prove that there
will be a limiting velocity, i.e., that there will be a limit to how fast you
will fall. There will be a limit to how fast you will fall even if you are
not wearing a parachute but the limit will be a much larger number. To see
a description of this complete with an animation go to free fall compared to using a parachute.
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If you really want to investigate this topic in depth
go to Drag
Forces on Solid Objects. This learning module from RPI includes an
analysis of skydiving and has a cute skydiver animation as well as other
instructional animations and short videos. |
| Here are three little animations
depicting projectile motion with air resistance taken as proportional to
velocity, i.e., R = kv. In each animation the initial height is 0
and the initial velocity is 100 ft/sec. In the first
animation the launch angle is 45o and you will see the path
of the projectile in blue with air resistance taken to be 0 and the
changing path of the projectile in red as air resistance varies in value
from 0.02 to 0.27. In the second
animation you will see the changing path of the projectile in blue
with air resistance taken to be 0 and the changing path of the projectile
in red as air resistance is taken to be 0.16. The path changes will
be due to the launch angle varying from 30o to 45o.
Notice that without air resistance the range of the projectile continues
to increase all the way up to a launch angle of 45o but with
air resistance the range of the projectile reaches a maximum before the
launch angle reaches 45o and then decreases. To see this
more clearly look at the third animation
in which the fixed path in blue corresponds to a launch angle of 45o with
air resistance set at 0.16 and initial velocity 100 ft/sec. and the
variable path in red corresponds to the same initial velocity and air
resistance but with the launch angle varying from 30o to 45o.
You will see the range of the path shown in red creep past that in blue
before the launch angle value reaches 45o.
More material on projectile motion can be found
in my Examples For Exam
II from my Calculus III course.
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