Calculus 10 Extra Topic

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10 extra topic Projectile Motion
Fort Pulaski, GA
Photo by Vickie Kelly, 2002
Greg Kelly, Hanford High School, Richland,
Washington
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One early use of calculus was to study projectile
motion.
In this section we assume ideal projectile motion
Constant force of gravity in a downward direction
Flat surface
No air resistance (usually)
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We assume that the projectile is launched from
the origin at time t 0 with initial velocity vo.
The initial position is
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Newtons second law of motion
Vertical acceleration
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Newtons second law of motion
The force of gravity is
Force is in the downward direction
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Newtons second law of motion
The force of gravity is
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Newtons second law of motion
The force of gravity is
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Initial conditions
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Vector equation for ideal projectile motion
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Vector equation for ideal projectile motion
Parametric equations for ideal projectile motion
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Example 1 A projectile is fired at 60o and 500
m/sec. Where will it be 10 seconds later?
The projectile will be 2.5 kilometers downrange
and at an altitude of 3.84 kilometers.
Note The speed of sound is 331.29 meters/sec Or
741.1 miles/hr at sea level.
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The maximum height of a projectile occurs when
the vertical velocity equals zero.
time at maximum height
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The maximum height of a projectile occurs when
the vertical velocity equals zero.
We can substitute this expression into the
formula for height to get the maximum height.
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maximum height
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When the height is zero
time at launch
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When the height is zero
time at launch
time at impact (flight time)
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If we take the expression for flight time and
substitute it into the equation for x, we can
find the range.
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If we take the expression for flight time and
substitute it into the equation for x, we can
find the range.
Range
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The range is maximum when
is maximum.
Range is maximum when the launch angle is 45o.
Range
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If we start with the parametric equations for
projectile motion, we can eliminate t to get y as
a function of x.
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If we start with the parametric equations for
projectile motion, we can eliminate t to get y as
a function of x.
This simplifies to
which is the equation of a parabola.
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If we start somewhere besides the origin, the
equations become
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Example 4
A baseball is hit from 3 feet above the ground
with an initial velocity of 152 ft/sec at an
angle of 20o from the horizontal. A gust of wind
adds a component of -8.8 ft/sec in the horizontal
direction to the initial velocity.
The parametric equations become
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These equations can be graphed on the TI-89 to
model the path of the ball
t2
Note that the calculator is in degrees.
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Using the trace function
Max height about 45 ft
Time about 3.3 sec
Distance traveled about 442 ft
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In real life, there are other forces on the
object. The most obvious is air resistance.
If the drag due to air resistance is proportional
to the velocity
(Drag is in the opposite direction as velocity.)
Equations for the motion of a projectile with
linear drag force are given on page 546.
You are not responsible for memorizing these
formulas.
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