Two Dimentional Motion

1. Projectile-Any object upon which the only force acting on is gravity!

2. Neglecting air • resistance- all objects • fall with the same • acceleration. • Two objects of different • weights, dropped from • the same height, hit • the ground at the same • time.

3. A horizontally • launched object will • reach the ground in • the same time as an • object dropped • vertically from rest. • The initial horizontal • velocity is irrelevant.

5. All projectiles follow a parabolic path

6. Horizontal Motion Constant, no acceleration. Vertical Motion Free fall, acceleration due to gravity.

7. A horizontally launched projectile The x-component of the velocity never changes

10. Which hits the ground first?

12. A Package Dropped from an airplane Initial vertical velocity equals zero

13. Vi=20 m/s The time of flight for all horizontally launched projectile Same as if it were dropped from rest

14. vi = 20 m/s d = 100 m 0 Calculate the time of flight for the projectile

15. Projectile launched from ground level at an angle

16. Projectile launched from ground level at an angle

18. Finding the initial vertical and horizontal components of the initial velocity given the total velocity, magnitude and direction!

19. A football is punted at 600 to the horizontal at 50 m/s, calculate the initial horizontal and vertical components of the velocity.

20. Given the initial horizontal and vertical velocities you can find the total velocity ( speed and direction.)

21. A motorcycle daredevil is attempting to jump across as many buses as possible. The takeoff ramp makes an angle of 180 above the horizontal, and the landing ramp is identical to the takeoff ramp. Each bus is 2.74 m wide. The cyclist leaves the ramp with a speed of 33.5 m/s. What is the maximum number of buses over which the cyclist can jump?

22. A projectile launched at an angle above the horizontal

23. Firing at Different Angles What launch angle(s) give the greatest range?

24. What launch angle(s) give the greatest range? 450 300 300 and 600 150 and 750

25. Range of a Projectile for vi = 50 m/s

26. Angles Summary • Complimentary angles give the same range • 450 gives the greatest range • 900 gives the greatest time of flight • 900 gives the greatest height

27. Air Resistance

28. Both car and ball have the same horizontal velocity link

29. Two assumptions: • Free-fall acceleration • g is constant. • Air resistance is • negligible. • The path of a projectile is parabolic. • The initial velocity is vi • The launched angle is  • Velocity vector changes in magnitude and direction • Acceleration in y-direction (vertical) is g • Acceleration in x-direction (horizontal) is 0

30. Superposition of motion in x-direction and motion in y-direction Acceleration in x-direction is 0. Acceleration in y-direction is g. (Constant velocity) (Constant acceleration) The horizontal motion and vertical motion are independent of each other; that is, neither motion affects the other.

31. Relative Motion Moving frame of reference A boat heading due north crosses a river with a speed of 10.0 km/h. The water in the river has a speed of 5.0 km/h due east. • Determine the velocity of the boat. • If the river is 3.0 km wide how long does it take to cross it?

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33. Displacement vs. time 0 d (m) If position is recorded, a parabolic d - t graph is formed. d - displacement t- time t (s)

34. Velocity vs. Time 0 0 v (m/s) d(m) t (s) t (s) If the velocity is recorded, a v-tlinear graph is formed. v - velocity t - time

35. 0 Velocity vs. Time v (m/s) Ball is moving up, positive velocity t (s) highest point, the velocity is zero. Ball is moving down negative velocity

36. 0 Velocity vs. Time v (m/s) Speed of the ball is reducing gradually. negative acceleration t (s) speed is zero, at rest. speed is increasing

37. v (m/s) t (s) 0 Areas in a Velocity vs. Time graph Area A = (+ displacement) A B Area B = (- displacement) The relationship between A and B ?

38. v (m/s) t (s) 0 Velocity vs. Time graph The slope is : g = the acceleration due to gravity , about -10 m/s2 What is the slope? v Slope = acceleration!

39. A ball tossed upward has initial velocity components 30 m/s vertical, and 5 m/s horizontal. The position of the ball is shown at 1-second intervals. Air resistance is negligible, and g = 10 m/s².

40. Motion of a ball falling off of a table. Which is the correct path?

41. A stone was thrown straight upward at t =0 with 20 m/s initial velocity on the roof of a 50 m high building • Find the time the stone reaches at maximum height (v=0) • Find the maximum height • Find the time the stone reaches its original height • Find the velocity of the stone when it reaches its original height • Find the velocity and position of the stone at t=5.0s

42. Find the horizontal and vertical components of the initial velocity. Carl Lewis at the 1992 Olympics in Barcelona, Spain, Lewis won gold medals for the long jump (28 feet 5.5 inches), this resulted from an initial velocity of 9.5 m/s at an angle of 40 degrees to the horizontal. Adam Vinatieri the New England Patriot kicker launches a kick at an angle of 300 at a velocity of 30 m/s Michelle Wie “The Big Wieasy” hits a nine iron with a velocity of 25 m/s at an angle of 600 to the horizontal.

43. A melon truck brakes right before a ravine and looses a few melons. • The melons skit over the edge with an initial velocity of v = 10 m/s • Determine the velocity just before in hits the ground • Calculate the velocity and the speed of the melon at t = 5.0 s • Determine the time it takes to hit the ground • Graph the path of a melon

44. Wiley Coyote has missed the elusive roadrunner once again. This time, he leaves the edge of the cliff at 50 m/s horizontal velocity. If the canyon is 100 m deep, how far from the edge of the cliff does the coyote land?

45. Throwing a ball out from a rooftop • A ball is thrown horizontally from • the Empire State building at 20 m/s • and lands 34.2 seconds later. Find: • The distance the ball lands from the • base of the Empire State building. • The final velocity of the ball. initial horizontal motion v e r t i c a l m o t i o n final

46. Bear Canyon Jump Malik is stranded on the wrong side of a 95 meter wide canyon with a grizzly bear. In a desperate attempt to escape from the bear he runs right off the edge of a cliff. He leaves at a velocity of 10 m/s horizontally. The canyon was 150 m deep, how long does he take to hit the ground? Or, does he make it to the other side? vi 150 m 95 m

47. A rescue plane flies at 40 m/s and an elevation of 100 m toward a point directly over a hiking accident victim. The pilot wants to release a rescue capsule so that it hits the ground very close to the victim. How far in front of the victim should he release the package?

48. A banana thrown at a monkey with no gravity

49. A banana thrown straight at the monkey

50. A banana thrown at a slow speed at the monkey