What is a Vector? A vector is a mathematical object which describes magnitude and direction

Transcription

1 What is a Vector? A vector is a mathematical object which describes magnitude and direction We frequently use vectors when solving problems in Physics Example: Change in position (displacement) Velocity

2 Notation We denote a vector by drawing an arrow above the variable For Example: Δ x, v, a, F When talking about vectors, we call normal numbers scalars

3 Direction In one dimensional space, we may use a sign to denote the vector s direction For this sign to have any meaning, we must define the positive and negative directions For Example + Up + W N S E Down

4 Displacement Displacement is a vector Describes the change in an object s position Written as: Δ x = x f x 0

5 Displacement Δ x = x f x 0 Δ is the Greek capitol letter Delta Δ denotes the change in a quantity

6 Displacement Δ x = x f x 0 x f is the final position x 0 is the initial position Remember: Δ x is final minus initial

7 Displacement Why is displacement final minus initial? Consider a person walking 5 m in the positive direction Since the person is walking in the positive direction, we want Δ x to be positive

8 Displacement x 0 0 x f meters Δ x = x f x 0 = 5 m 0 m = +5 m

9 Distance vs. Displacement Displacement does not depend on the path taken, only the initial and final positions All three paths have the same displacement

10 Distance vs. Displacement Distance is the total length of the path taken Distance does depend on the path An odometer measures distance Which path has the shortest distance?

11 Distance vs. Displacement If a student drives 4 miles from home to school, and, at the end of the day, 4 miles from school back home. What was his displacement? What was the total distance traveled?

12 Distance vs. Displacement meters What was his displacement? What was the total distance traveled?

13 Distance vs. Displacement meters What was his displacement? What was the total distance traveled?

14 Velocity is a vector Denoted as v Velocity Describes how fast and in what direction an object is moving

15 Velocity In this class we will work mostly with average velocity. The average velocity is given by: v avg = x f x 0 = Δ x t f t 0 Δt v avg always points in the same direction as Δ x Note: the textbook denotes v avg as v

16 Velocity vs. Speed Speed vs. velocity similar to distance vs. displacement Speed is a scalar The average speed is defined as: speed = total distance time

17 Velocity - Example Between first and second period, you walk m West to your second period class. It takes you seconds to get to class. What was your average velocity in m s?

18 Velocity - Example Your third period classroom is 150 m away from your second period classroom. You have four minutes between classes. Assuming you walk in a direct path, at this same speed, will you make it to class on time?

19 Velocity - Example In the morning, you drive 5.0 km East to get to school. It takes 20.0 minutes to get to school. What is your velocity in m s?

20 Velocity - Example In the afternoon, you drive 5.0 km West to go back home. It takes 10.0 minutes to get home. What is your average velocity in m s? What was the average velocity of the entire trip?

21 Velocity - Example Notice that : The total trip, v avg = 0 m s Therefore, v avg +4.2 m s 8.3 m s v avg m s 8.3 m s

22 Velocity - Example One morning you drive 8.0 km East to McDonald's for breakfast. It takes you s to get to there from your house. You then drive 2.0 km West to Central Catholic; it takes s.

23 Velocity - Example s West s East km What was the average velocity in m s to McDonalds? during the trip

24 Velocity - Example s West s East km What was the average velocity in m during the trip s from McDonalds to Central Catholic?

25 Velocity - Example s West s East km What was the average velocity in m s trip? of the entire

26 Velocity - Example s West s East km What was the average speed in m s trip? of the entire

27 Acceleration Acceleration describes the rate at which v(t) changes Like velocity & displacement, acceleration is a vector quantity a avg = Δ v Δt

28 Acceleration Recall that v is a vector Has magnitude & direction a 0 m/s 2 when v changes, this will occur whenever: The magnitude of v changes (speed of the object changes) The motion changes direction

29 Acceleration Even if an object is traveling at a constant speed, if the direction of the motion changes, the object is accelerating.

30 Acceleration vs. Deceleration Deceleration speed is decreasing Deceleration occurs when the acceleration opposes the motion Deceleration is not the same as negative acceleration

31 Acceleration vs. Deceleration Example: A falling apple experiences a + Up negative acceleration because it is falling downward. Down a However, the apple is not decelerating.

32 Acceleration Example While driving you are stopped at a red light. When the light turns green you begin acceleration. 30 seconds later you have a velocity of +25 m. What is s your acceleration?

33 Acceleration Example While driving +25 m, you see a red light. s You begin to decelerate at a rate 1.5 m s 2. How long does it take for the car to come to rest?

34 When a = cosnt., a avg = a v avg = 1 2 v f + v 0 Using this along with: a = Δv Δt v avg = Δ x Δt We can derive two important equations in kinematics Constant Acceleration

35 Constant Acceleration - Example You are on the entrance ramp to I-79. You start out moving at +25 m s. You accelerate at a constant rate of +5 m s seconds later you merge. How far down the ramp did you drive?

36 v 0 = +25 m s a = 5 m s 2 Δt = 10 s Δx =?

37 Constant Acceleration In the previous example we found a useful formula: Δ x = v 0 Δt a Δt 2 Depends on Δx, a, Δt, v 0 Does not depend on v f

38 Constant Acceleration You move onto the exit ramp on I-70 with an initial velocity of +60 m s. You decelerate at a constant rate of 5 m s 2 to a final velocity of +20 m s. How far does the car travel before reaching this final velocity?

39 v 0 = +60 m s v f = +20 m s a = 5 m s 2 Δ x =?

40 Constant Acceleration In the previous example we found a useful formula Δ x = v f 2 v 0 2 2a Which we can rewrite as: Depends on Δx,a, v f, v 0 Does not depend on Δt

41 Free Fallin Objects in free-fall accelerate towards the ground with a constant acceleration g = 9.80 m ft s2 or 32.2 s 2 Typically describe height using y and Δy Usually we say the positive y-direction is up, and the negative y-direction is down

42 Free-Fall Example A rocket is launched from the top of a 100 m building. The rocket is launched vertically upwards with an initial speed of 30 m s. What is the speed of the rocket, when it hits the ground at the bottom of the building?

43 v 0 = +30 m s a = 9.80 m s 2 Δy = 100 m v f =?

44 Free-Fall Symmetry Free-fall problems exhibit a natural symmetry y t return = 2 t top t top t t return

45 Free-Fall Symmetry y -v 0 At the top of the trajectory, v changes sign v = 0 m at the s top of the trajectory v 0 v t t

46 -v 0 Free-Fall Symmetry y When an object returns to its initial height, v = v 0 v 0 v t t

47 Free-Fall Symmetry How long does it take for an object to reach the top of its trajectory when thrown upwards with an initial velocity v 0?

48 Free-Fall Symmetry How long does it take for the object to return to its initial height?

49 Free-Fall Example Consider the rocket from the previous example. Recall that the rocket was launched vertically upwards from a 100 m building. The rocket left the ground with an initial velocity of +30 m s, and its engines burn up almost immediately. What was the maximum height achieved by this rocket?

50 v 0 = +30 m s v f = 0 m s a = 9.80 m s 2 y max =?

51 Free-Fall Example At the same moment the rocket is launched upwards, at 30 m, an identical s rocket is launched directly downwards at the same speed. How much longer does it take the rocket which was launched upwards to hit the ground?

52 Free-Fall Example

53 v 0 = +30 m s Δt =? a = 9.80 m s 2

54 Recall that: Graphical Analysis y slope = Δy Δx = y f y 0 x f x 0 Δy Δx x

55 Graphical Analysis Constant Velocity For position vs. time: x slope = Δx Δt = v avg Δx Δt t

56 Instantaneous Velocity In most situations, an object s velocity changes in time Instantaneous velocity is the velocity at a particular instant in time To calculate instantaneous velocity, we calculate v avg over a extremely short period of time

57 Graphical Analysis Changing Velocity x v avg = Δx Δt Δx Δt t

58 Graphical Analysis Instantaneous Velocity x slope = v(t) t t

59 Graphical Analysis Instantaneous Velocity v(t) is the slope of the tangent line at t v(t) is graphically understood as the steepness of the x(t) vs t graph.

60 Graphical Analysis Constant Acceleration For velocity vs. time: v slope = Δv Δt = a avg Δv Δt t

61 Graphical Analysis x t What does v(t) look like?

62 x t v t

63 Graphical Analysis Just like the derivative is graphically linked to the steepness of a graph, the integral also has a graphical interpretation. Consider an object traveling at the constant velocity. The displacement is given by: Δx = v Δt

64 Graphical Analysis Recall that the product of two values can be visualized as the area of a rectangle. v Area = v Δt = Δx Δt

65 Graphical Analysis When given a graph of v vs. t, we can visualize Δx as the area under v(t) over a time interval, Δt v v Δt t

66 Graphical Analysis If velocity is not constant, we can: Break a time interval Δt, into smaller parts, of length δt Assume v(t) stays nearly constant for a short time, δt Use δx = v t δt to calculate the displacements that occur during each δt We can visualize each δt as a rectangular area

67 Velocity Graphical Analysis Time

68 Velocity Graphical Analysis Time

69 Velocity Graphical Analysis Time

70 Graphical Analysis In general: Δx = Area Where Area means the area between v(t) and the time-axis.

71 Graphical Analysis When considering graphs of x(t) and v t remember: x(t) is the area under v(t) v(t) is the slope of x(t)

72 t v Graphical Analysis t Δx

73 Graphical Analysis When v(t) is below the horizontal axis, Δx is negative.

74 Graphical Analysis v m s 5 m s t (s)

75 Graphical Analysis v m s +4 m s t (s) 4 m s