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10.4 Projectile Motion Fort Pulaski, GA Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington 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

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10.4 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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Newton’s second law of motion: time

Vertical acceleration


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Newton’s second law of motion: time

The force of gravity is:

Force is in the downward direction


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Newton’s second law of motion: time

The force of gravity is:


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Newton’s second law of motion: time

The force of gravity is:



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Vector equation time for ideal projectile motion:


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Vector equation for ideal projectile motion: time

Parametric equations for ideal projectile motion:


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Example 1: time

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 time maximum height of a projectile occurs when the vertical velocity equals zero.

time at maximum height


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The time 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 time

height


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When the time height is zero:

time at launch:


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When the time 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 into the equation for x, we can find the range.

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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Example 4: become:

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 path of the ball:

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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