What is Projectile Motion?

1. 0 What is Projectile Motion?

2. 0 Instructional Objectives: • Students will be able to: • Define Projectile Motion • Distinguish between the different types of projectile motion • Apply the concept to a ball rolling off a table and measure its velocity

3. 0 Projectile Motion • Same requirements as free fall, but projectile motion is two-dimensional motion of an object • Vertical direction • Horizontal direction

4. 0 Concepts of Projectile Motion • Horizontal • Motion of a ball rolling freely along a level surface • Horizontal velocity is ALWAYS constant • Nothing pushes or pulls on it in the horizontal direction to cause it to accelerate • Vertical • Motion of a freely falling object • Force due to gravity • Vertical component of velocity changes with time • Parabolic • Path traced by an object accelerating only in the vertical direction while moving at constant horizontal velocity

5. 0 Parabola

6. 0 Examples of Projectile Motion • Launching a Cannon ball

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8. 0 Factors Affecting Projectile Motion • What two factors would affect projectile motion? • Angle • Initial velocity Initial Velocity Angle

9. 0 • The range R is the horizontal distance the projectile has traveled when it returns to its launch height.

10. 0 Projectile Motion Solutions PROBLEM-SOLVING STRATEGY Projectile motion problems MODELMake simplifying assumptions. VISUALIZE Use a pictorial representation. Establish a coordinate system with the x-axis horizontal and the y-axis vertical. Show important points in the motion on a sketch. Define symbols and identify what the problem is trying to find. SOLVE The acceleration is known: and thus the problem becomes one of kinematics. The kinematic equations are (THESE ARE NOT NEW – THEY ARE JUST OUR 1-D EQUATIONS): is the same for the horizontal and vertical components of the motion. Find from one component, then use that value for the other component. ASSESS Check that your result has the correct units, is reasonable, and answers the question.

11. 0 Horizontal motion • No acceleration

12. 0 Vertical motion (Equations of Motion ): 1 ) 2 ) 3 )

13. 0 For a particular range less than the maximum and for a particular launch velocity, two different launch angles will give that range. The two angles add to give 900. 450 gives the maximum range.

14. 0 Example Problem 1 An object is fired from the ground at 100 meters per second at an angle of 30 degrees with the horizontal • Calculate the horizontal and vertical components of the initial velocity • How long does it take to reach highest point? • How far does the object travel in the horizontal direction?

15. Example: A Home Run A baseball is hit so that it leaves the bat making a 300 angle with the ground. It crosses a low fence at the boundary of the ballpark 100 m from home plate at the same height that it was struck. (Neglect air resistance.) What was its velocity as it left the bat?

16. 0 Sample Problem 2 • In Fig. 4-15, a rescue plane flies at 198 km/h (= 55.0 m/s) and a constant elevation of 500 m toward a point directly over a boating accident victim struggling in the water. The pilot wants to release a rescue capsule so that it hits the water very close to the victim.

17. 0 (a)  What should be the angle of the pilot's line of sight to the victim when the release is made? Solving for t, we find t = ± 10.1 s (take the positive root). Solution

18. 55 m/s 61° 99 m/s 113 m/s 0 (b)  As the capsule reaches the water, what is its velocity as a magnitude and an angle? When the capsule reaches the water,

19. 0 Sample Problem 3Figure 4-17 illustrates the flight of Emanuel Zacchini over three Ferris wheels, located as shown and each 18 m high. Zacchini is launched with speed v0 = 26.5 m/s, at an angle = 53° up from the horizontal and with an initial height of 3.0 m above the ground. The net in which he is to land is at the same height. • (a)  Does he clear the first Ferris wheel?

20. 0 SOLUTION • The equation of trajectory when x0 = 0 and y0 = 0 is given by : Solving for y when x = 23m gives Since he begins 3m off the ground, he clears the Ferris wheel by (23.3 – 18) = 5.3 m

21. 0 (b)  If he reaches his maximum height when he is over the middle Ferris wheel, what is his clearance above it? • SOLUTION:  At maximum height, vy is 0. Therefore, and he clears the middle Ferris wheel by (22.9 + 3.0 -18) m =7.9 m

22. 0 (c)  How far from the cannon should the center of the net be positioned? • SOLUTION: 

23. 0 FAST-MOVING PROJECTILES - SATELLITES • Let’s throw stones horizontally with ever increasing velocity. • The Earth’s curvature is 16 ft for every 5 miles (4.9 m for 8 km).

24. 5 miles 16 ft 16 ft 0 Throw an object faster, faster, faster. Curvature of Earth We draw in each trajectory for 1 second

25. Vescape = 7 mi/s = 11 km/s Vcircle = 5 mi/s = 8 km/s 0 Ellipse Hyperbola Parabola Ellipse Ellipse - Circle Ellipse Ellipse

26. 0 A rock is thrown upward at an angle. What happens to the horizontal component of its velocity as it rises? (Neglect air resistance.) (a) it decreases (b) it increases (c) it remains the same

27. 0 In baseball which path would a home run most closely approximate? (Neglect air resistance.) (a) hyperbolic (b) parabolic (c) ellipse

28. 0 A projectile is launched upward at an angle of 75° from the horizontal and strikes the ground a certain distance down range. What other angle of launch at the same launch speed would produce the same distance? (Neglect air resistance.) (a) 45° (b) 15° (c) 25°

29. 0 A horizontally traveling car drives off of a cliff next to the ocean. At the same time that the car leaves the cliff a bystander drops his camera. Which hits the ocean first? (Neglect air resistance.) (a) car (b) camera (c) they both hit at the same time