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# 10.4 Projectile Motion - PowerPoint PPT Presentation

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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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

Flat surface

No air resistance (usually)

We assume that the projectile is launched from the origin at time t=0 with initial velocity vo.

The initial position is:

Vertical acceleration

The force of gravity is:

Force is in the downward direction

The force of gravity is:

The force of gravity is:

Vector equation time for ideal projectile motion:

Parametric equations for ideal projectile motion:

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.

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

time at maximum height

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.

maximum time

height

When the time height is zero:

time at launch:

When the time height is zero:

time at launch:

time at impact

(flight time)

If we take the expression for flight time and substitute it into the equation for x, we can find the range.

If we take the expression for flight time and substitute it into the equation for x, we can find the range.

Range

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

If we start with the parametric equations for projectile motion, we can eliminate t to get y as a function of x.

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.

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:

These equations can be graphed on the TI-89 to model the path of the ball:

t2

Note that the calculator is in degrees.

Using path of the ball:

the

trace

function:

Time

3.3 sec