1. 10.4 Projectile Motion Fort Pulaski, GA Mackinaw Island, Michigan
Photo by Vickie Kelly, 2002
Greg Kelly, Hanford High School, Richland, Washington

2. 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)

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

4. Newton’s second law of motion:
Vertical acceleration

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

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

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

8. Initial conditions:

9. Vector equation for ideal projectile motion:

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

11. 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 meters/sec
Or miles/hr at sea level.

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

13. 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.

15. maximum
height

16. When the height is zero:
time at launch:

17. When the height is zero:
time at launch:
time at impact
(flight time)

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

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

20. The range is maximum when
Range is maximum when the launch angle is 45o.
Range

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

22. 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.

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

24. 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:

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

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

28. 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