MOTION IN TWO DIMENSIONS

1. MOTION IN TWO DIMENSIONS Some diagrams and simulations from www.physicsclassroom.com

3. Projectile Motion Newton's laws help to explain the motion (and specifically, the changes in the state of motion) of objects which are either at rest or moving in 1-dimension. One of the most common example of an object which is moving in two-dimensions is a projectile. PROJECTILE: is an object upon which the only force acting is gravity.

5. EXAMPLES OF PROJECTILES an object dropped from rest is a projectile an object which is thrown vertically upwards is also a projectile an object which is thrown upwards at an angle is also a projectile

6. MISCONCEPTION: Some people think that a projectile must have a force acting upward upon it in order for it to climb. This is not true. A force is not required to keep an object in motion. A force is only required to maintain acceleration. The path of a projectile is not that complicated. Many projectiles not only undergo a vertical motion, but also undergo a horizontal motion. The horizontal and vertical components of the projectile’s motion are independent of each other.

12. Vector Combination

13. Horizontal Motion Vertical Motion Forces No Yes The force of gravity acts downward Acceleration No Yes "g" is downward at ~ -9.8 m/s/s Velocity Constant Changing (by ~ 9.8 m/s each second)

14. WHAT IF THE PROJECTILE IS SHOT UPWARD?

16. Now assume gravity is turned on The projectile would travel with a parabolic trajectory. The downward force of gravity will act upon the cannonball to cause the same vertical motion as before - a downward acceleration. The cannonball falls the same amount of distance in every second as it did when it was merely dropped from rest.

18. Projection Angles With the same initial speed but different projection angles, a projectile will reach different altitudes (height above the ground) and different ranges (distances traveled horizontally). However, the same range can be obtained from two different angles, symmetrically around a maximum of 45˚, as shown in the graph.

21. Symmetry The path of a projectile is symmetrical… It rises to its maximum height in the same time it takes to fall from that height to the ground. Because acceleration is the same all of the time, the speed it loses while going up is the same as the speed it gains while falling. Therefore the speeds are the same at equal distances from the maximum height, where the vertical speed is zero.

22. Question: At the instant a horizontally held rifle is fired over level ground, a bullet held at the side of the rifle is released and drops to the ground. Ignoring air resistance, which bullet strikes the ground first? They strike at the same time.

23. Question: A projectile is launched at an angle into the air. If air resistance is negligible, what is the acceleration of its vertical component of motion? Of its horizontal component of motion? Vertical -9.8 m/s2 Horizontal 0 m/s2

24. Question: At what part of its trajectory does a projectile have minimum speed? At maximum height.

25. Question: A ball tossed into the air undergoes acceleration while it follows a parabolic path. When the sun is directly overhead, does the shadow of the ball across the field also accelerate? NO

26. Question: How can an object be moving upward if the only force acting upon it is gravity? Newton’s first law

27. Question: What launch angle maximizes the range (horizontal distance)? 45°

28. Question: • What launch angle maximizes the height reached? 90°

29. Question: How does the time spent in the air depend on the launch angle? Time is maximized at 90°

30. Question: Compare what happens at complementary launch angles. Same range (horizontal displacement)

31. Question: What happens to the trajectory when the mass of the projectile is changed? Nothing

32. Question: • Compare the trajectories of a projectile that is under the influence of gravity and one that is not? NO Gravity (straight line) Gravity (parabolic trajectory)

33. Simulation #1

34. Simulation #2

35. Do Q’s 1-4 of Handout Study Guide 7.1 Q’s 4-8 of Handout Q’s 1-8 pg 536 & 537 (pdf 67)

36. Periodic Motion

37. Periodic Motion Projectile motion is two-dimensional, but it does not repeat. Projectiles do not move all along their trajectories more than once. Periodic motion can also be an example of 2 dimensional motion however it involves motion that repeats itself at regular intervals. Examples of periodic motion are a yo-yo being swung horizontally overhead, an object bouncing on a spring or the pendulum of a clock.

38. Circular motion An object that moves in a circle at constant speed is said to experience UNIFORM CIRCULAR MOTION. Recall that velocity is a vector quantity so it has both magnitude and direction. With circular motion an object may have a constant speed but a direction that is constantly changing. If this is the case then the object is said to be undergoing centripetal acceleration.

39. Centripetal acceleration: centripetal means center seeking, or acceleration in the direction of the center of the circle and can be found using the formula Units? Centripetal acceleration always points towards the center of the circle and is directly proportional to the square of the speed and inversely proportional to the radius of the circle.

40. The velocity of an object undergoing circular motion can be found using the following formula, Where the distance traveled is equal to the circumference of the circle, and the total time is one period. PERIOD (T) - the length of time needed to complete one cycle of motion.

41. by combining the two above equations we get a second formula for ac

42. Newton’s second law tells us that an object does not accelerate unless there is a force that acts on it. For circular motion this force is called a centripetal force and is also directed radially inwards.

43. This diagram shows the centripetal force acting in the same direction as the acceleration. If you remove the centripetal force, the object will not continue moving in a circular path. Examples: merry-go-round in the park, tilt-a-whirl, Nascar banking on the turn at Talledega

44. Example: A 0.013 kg rubber stopper is attached to a 0.93 m length of string. The stopper is swung in a horizontal circle, making one revolution in 1.18 seconds. a) Find the speed of the stopper. b) Find its centripetal acceleration. c) Find the force that the string exerts on it. a) 5.0 m/s, b) 26 m/s2, c) 0.34 N

45. Do Ring around the collar Q’s 9-15 Handout Q’s 15- 19 pg 559 (pdf 68) omit # 17

46. Simple Harmonic Motion Simple harmonic motion is another example of periodic motion. The key thing with simple harmonic motion is that there must be a restoring force that causes the object to return to the equilibrium position.

47. The restoring force must vary linearly with respect to displacement. • ie: small displacement, small force • large displacement, large force The maximum displacement is called amplitude. Amplitude: the maximum distance that the object moves from its equilibrium position.

48. An example of a type of simple harmonic motion would be that of a pendulum. Note that the restoring force here is the tangential component of the weight and that it increases with amplitude.

49. Similar to the pendulum, the spring is yet another example of SHM where the restoring force varies linearly with respect to displacement and is always in the direction of the equilibrium position.

50. Pendulum Spring The period of a simple pendulum and that of a spring can be found by the following formulas,