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EXAMPLES FOR EXAM II Section 12.3
| If it is not already on your hard drive, you will need to download
the free DPGraph Viewer to view some of
the pictures linked to on this page. |
QuickTime
7 free download. |
Click here to see a
path animated with its position function vectors, velocity function vectors, and
acceleration function vectors. Here is a
second, similar demonstration. Here is a third
such demonstration involving vectors in space. This fourth demonstrates a
spiral in space. Here is
one demonstrating a Lissajous figure and a 3-D
demo combining Lissajous and spiral.
Projectile Motion Example
A projectile is launched from ground level at an angle of 45o
with the horizontal and with an initial velocity of 64 feet per second. A
television camera is located in the plane of the path of the projectile 50 feet
behind the launch site.
Parametric equations for the path of the projectile in
terms of the parameter t representing time are
Some of the basic things we can easily compute are the
maximum height attained by the projectile and the range of the projectile.
The graph in red below represents the path of the
projectile and the blue point moving along the graph in red represents
the projectile. The length of the vertical blue line segment at x = -50
represents the measure of angle a (the angle the camera makes with the
horizontal) in degrees
animation of the
projectile motion and changing angle a
The angle a that the camera makes with the
horizontal is given by
Below is a graph of the measure of angle a in
degrees as a function of time (t).
Notice that a is not a maximum at the same time
that y is a maximum.
| Animation For Section 12.3, Example
6
A baseball is hit 3 feet above the ground at 100
feet per second and at an angle of 45o with respect to the
ground. Find the maximum height reached by the baseball. Will
it clear a 10 foot high fence located 300 feet from home plate? The solution can be found in your textbook.
Click on the picture at the right to see the animation. Quicktime
Animation Quicktime
Animation Extended
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| Section 12.3 #34 (similar--yards
changed to feet) The quarterback of a football team releases a
pass at the height of 7 feet above the playing field, and the football is
caught by a receiver 30 feet
directly downfield at a height of 4 feet. The pass is released at an
angle of 35o with the horizontal.
Click on the picture to see an animation.
In my animation the receiver is 20 feet
directly downfield when the
quarterback releases the football.
Quicktime
version
(a) Find the speed of the football when it
is released.
(b) Find the maximum height of the
football.
(c) Find the time the receiver has to reach
the proper position after the quarterback releases the football.
The receiver has approximately 1.2249 seconds to
reach the proper position after the quarterback releases the football.
Highlights of the Solution to Section 12.3 #34 (The difference is that the 30 feet
is 30 yards.)
Animation
for the original problem
Quicktime Version
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| Section 12.3 #36
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| The Path of the Bomb with Air
Resistance
The picture below shows the path the bomb would
take if we do not neglect air resistance but rather take air resistance as
proportional to velocity (of the bomb), i.e., r = kv. The time
interval is the length of time it takes the bomb to hit the ground
neglecting air resistance (approximately 43.3 seconds). The successive values for r in the graph,
from left to right, are: 0.1, 0.08, 0.06, 0.04, 0.02, 0.01, 0. Click
on the picture below to see a similar picture that also includes the paths
corresponding to r = 0.4 and r =0.2. The bomb falls very slowly if r
= 0.4. |
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| The graph
below is a graph of the endpoints of paths such as those shown above as r
goes from 0.000001 to 3. |
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Lawn Sprinkler
Here is an example of the lawn sprinkler problem found
in the exercises (61) for Section 3.1 (It is a Calculus III problem.). In
the example here the speed of the water
is 16 ft/sec so the distance the water travels horizontally is given by
and the path the water takes through the air is given
by
Can you see why (neglecting air resistance)? Answer
Click here to see
an animation for this problem and click here
for an animation with scales. Would this sort of lawn sprinkler water the
lawn uniformly? Answer For more
information on the "calculus of lawn sprinklers" see the article
"Design of an Oscillating Sprinkler" by Bart Braden in Mathematics
Magazine. You can view the article at matharticles.com.
| EC
The picture at the right shows the paths of a
projectile launched from sea level with an initial speed of 48
ft/sec. The projectile has been launched in the direction of a gully
whose flat bottom is 192 feet lower than the spot from which the
projectile was launched. The blue
path corresponds to a launch angle of 60 degrees. The red
path corresponds to a launch angle of 45 degrees. The green
path corresponds to the launch angle that maximizes the horizontal
distance traveled by the projectile. The black
path corresponds to a launch angle of 0 degrees. The extra credit is to
approximate the radian measure of the launch angle for the green path (6
significant digits), give the approximate degree measure (4 significant
digits), and approximate the horizontal distance traveled (6 significant
digits) for the green path. Air resistance is being ignored.
Click on the picture to see an animation showing the path of the
projectile as the launch angle varies from zero to almost ninety
degrees. Quicktime version of
the animation |
Maple
Version of the Animation Quicktime
Maple Version
Maple
Worksheet for the Maple Animation
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