1. x is the magnitude of a position v is the magnitude of the velocity, sometimes speed a is the magnitude of acceleration t is time Δ represents a change, an interval. Δx is displacement, and can be also called “d” sometimes Δv is a change in speed Variables used in kinematics: These will be vectors, they need a direction. These will be just numbers, or scalars.

2. Distance or displacement? This is a scalar. It is a number, a distance, doesn’t matter the direction of the trajectory. This is a vector. It is a number AND a direction. There are special rules to operate with vectors. It is not the same as a simple number.

3. What ’ s the difference between speed and velocity? Speed is a scalar. Scalars have only a magnitude (just a number). For example: constant speed of a car at 60 km/h. Velocity is a vector. Vectors have a magnitude AND a direction. For example: velocity of an airplane traveling 200 km/h Northward. Speed can be the magnitude of the velocity. 60 km/h

4. A car goes around a curve at constant speed. Is the car’s velocity changing? a)Yes b)No At position A, the car has the velocity indicated by the arrow (vector) v1. At position B, the car has the velocity indicated by the arrow (vector) v2, with the same magnitude (speed) but a different direction.

5. Our frame of reference (it is a necessary convention) + – – + +3+4 -2 -3 +4 -3-5

9. Quick Quizzes Answer True or False: 1) A car must always have an acceleration in the same direction as its velocity. 2) It’s possible for a slowing car to have a positive acceleration. 3) An object with constant nonzero acceleration can never stop and remain at rest. F T T

10. 000 11020 22040 33060 44080 550 100 time x car1 x car2 __ (s) (m) (m) This is a table showing the positions of car 1 and car 2 for a motion in 5 seconds. Which one moves further away in the same time? Which one has the greatest average speed? x (m) t (s) Displacement x time 0 Let’s plot this on the blackboard.

11. EQUATION OF A STRAIGHT LINE m is the slope b is the y-intercept (the value of y when x=0) b y x ΔyΔy ΔxΔx

12. m is the slope b is the y-intercept (the value of y when x=0) For the equation What is the slope? What is the y-intercept? For the equation What is the slope? What is the y-intercept?

13. What does a car’s speedometer measure? a)Average speed b)Instantaneous speed c)Average velocity d)Instantaneous velocity A speedometer measures instantaneous speed.

14. Which quantity is the highway patrol more interested in? a)Average speed b)Instantaneous speed The speed limit indicates the maximum legal instantaneous speed. To estimate the time a trip may take, you want to use average speed.

15. Passing lane Watch the red car Watch the blue car POSITION versus TIME http://www.physicsclassroom.com In a graph POSITION versus TIME, the SLOPE indicates VELOCITY. A steeper slope indicates greater velocity.

16. Passing lane Watch the red car Watch the blue car VELOCITY versus TIME In a graph VELOCITY versus TIME, constant velocity is a straight line. http://www.physicsclassroom.com

17. In general, Average ≠ Instantaneous

20. Is the instantaneous velocity at point A greater or less than that at point B? a)Greater than b)Less than c)The same as d)Unable to tell from this graph The instantaneous velocities can be compared by looking at their slopes. The steeper slope indicates the greater instantaneous velocity, so the velocity at A is greater.

22. In the graph shown, during which time interval is the acceleration greatest? a)Between 0 s and 2 s. b)Between 2 s and 4 s. c)Between 4 s and 8 s. d)The acceleration does not change. The acceleration is greatest between 2 s and 4 s. The velocity is changing fastest, and the graph has the greatest slope, during this interval.

23. At which point is the magnitude of the acceleration the greatest? a)Point A b)Point B c)Point C d)The acceleration does not change. The magnitude of the acceleration is greatest when the velocity is changing the fastest (has the greatest slope). This occurs at point A.

24. 2.5 CONSTANT ACCELERATION Important equations! Many applications of mechanics involve objects moving with constant acceleration. Constant “ a ” means that average ā = instantaneous a For convenience, let’s adopt v f = v and v i = v 0, t i = 0, t f = t so: or Because velocity is increasing or decreasing UNIFORMLY with time, we can express: average final Remember that: or 3 1 2 So now I will substitute inside, and then again inside the resulting equation. 2 31

25. Graph will be a straight line. Graph will be a straight line if a = 0, but it will be a parabola if a ≠ 0. Velocity depends on t Position depends on t and t 2 Graph v vs tGraph x vs t

26. From a graph: slope or area? Example: Slope Area Example:

27. What should you do if you have a velocity versus time graph and you want…. acceleration ? displacement ? AREAS v (m/s) t (s) SLOPE

28. During which time interval is the distance traveled by the car the greatest? a)Between 0 s and 2 s. b)Between 2 s and 4 s. c)Between 4 s and 6 s. d)It is the same for all time intervals. The distance traveled is greatest when the area under the velocity curve is greatest. This occurs between 2 s and 4 s, when the velocity is constant and a maximum.

29. Important: our equations can only be used in regions of CONSTANT ACCELERATION.

30. Constant velocity in the positive direction http://www.physicsclassroom.com

31. Constant velocity in the negative direction

32. CONSTANT POSITIVE ACCELERATION http://www.physicsclassroom.com

33. CONSTANT NEGATIVE ACCELERATION http://www.physicsclassroom.com

34. t v0v0 t 0 Velocity v Acceleration 0 a t v(t) a(t) = 0 Slope = 0 Position 0 x0x0 x x(t) Slope = v 0 Position 0 x t v0v0 t 0 Velocity v Acceleration 0 a a0a0 t x(t) a(t) v(t) Slope = 0 Slope = a 0 x0x0 Slope varies UNIFORM MOTION (velocity is constant) UNIFORM ACCELERATION (acceleration is constant)

35. Quick Quiz Parts (a), (b), and (c) of the figure below represent three graphs of the velocities of different objects moving in straight-line paths as functions of time. The possible accelerations of each object as functions of time are shown in parts (d), (e), and (f). Match each velocity vs. time graph with the acceleration vs. time graph that best describes the motion. 1.a and e, b and f, c and d 2.a and d, b and f, c and e 3.a and e, b and d, c and f

36. 0 24 68 10 Time (s) 0 1 2 3 4 5 Distance (m) Match this plot: 0 24 68 10 Time (s) 0 1 2 3 4 5 Velocity (m/s) (a) 0 24 68 10 Time (s) 0 1 2 3 4 5 Acceleration (m/s 2 ) (b) 0 24 68 10 Time (s) -2 0 2 Velocity (m/s) (c) 0 24 68 10 Time (s) 0 1 2 3 4 5 Velocity (m/s) (d) 0 24 68 10 Time (s) 0 1 2 3 4 5 Acceleration (m/s 2 ) (e) 1 3

37. 0 24 68 10 Time (s) 0 1 2 Acceleration (m/s 2 ) Match this plot: 24 68 10 Time (s) 0 4 Velocity (m/s) (b) 0 24 68 10 Time (s) 0 1 2 3 4 5 Velocity (m/s) (d) -2 0 24 68 10 Time (s) Distance (m) (a) 10 0 20 0 24 68 10 Time (s) 15 Distance (m) (c) 10 5 0 -5 20. 0 24 68 10 Time (s) 25 Distance (m) (e) 20 15 10 5 30 8 6 2 0 -2 30 40 50 60

41. Acceleration of gravity does not depend on the mass VACUUMAIR

42. Gravitational acceleration g does NOT depend on the weight of the object. Apollo 15 Moon walk in 1971. Commander David Scott

43. Equations governing falling objects and objects thrown upward

44. What is the ball’s acceleration at the top of its path (at t=2 s)? a)zero. b)+9.8 m/s c)-9.8 m/s d)+9.8 m/s 2 e)-9.8 m/s 2 Gravity does not “turn off” at the top! The ball’s velocity is still changing, as it changes from going up to going down. For a moment the velocity is zero, but the gravitational acceleration is a constant throughout the path.

45. t = 0 y = 0 v = + 20 m/s t = 1 s y = 15 m v = + 10 m/s t = 2 s y = 20 m v = 0 t = 3 s y = 15 m v = –10 m/s t = 4 s y = 0 m v = –20 m/s a = -g v v v v a a a a a Always: Let’s suppose now that the initial velocity of the ball upward is +35 m/s (instead of the +20 m/s as in the picture). What is the maximum height that the ball will reach? Assume that g = 10 m/s 2 Maximum height y max = 20 m Equations we can use: Answer: y max = 61 m

47. Quick Quizzes A tennis player on serve tosses a ball straight up. While the ball is in free fall, its acceleration 1. increases.2. decreases.3. increases and then decreases. 4. decreases and then increases.5. remains constant. A tennis player on serve tosses a ball straight up. As the tennis ball travels through the air, its speed 1. increases.2. decreases.3. increases and then decreases. 4. decreases and then increases.5. remains constant.