10.4 Projectile Motion Fort Pulaski, GA Mackinaw Island, Michigan 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:

Newton’s second law of motion: Vertical acceleration

Newton’s second law of motion: The force of gravity is: Force is in the downward direction

Newton’s second law of motion: The force of gravity is:

Newton’s second law of motion: The force of gravity is:

Initial conditions:

Vector equation for ideal projectile motion:

Vector equation for ideal projectile motion: Parametric equations for ideal projectile motion:

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.

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

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.

maximum height

When the height is zero: time at launch:

When the 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 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.

If we start somewhere besides the origin, the equations become:

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:

These equations can be graphed on the TI-89 to model the path of the ball: Note that the calculator is in degrees.

Using the trace function: Max height about 45 ft Time about 3.3 sec Distance traveled about 442 ft

In real life, there are other forces on the object 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. p