distance vs. displacement

Transcription

1 distance vs. displacement The distance traveled by an object is path dependent. The displacement is the change in position of an object. Example: A car travels 15 miles east and 5 miles west. distance traveled = 20 miles displacement = 10 miles

2 Average velocity The velocity is a vector with both magnitude & direction. The average velocity is the change in distance over some finite time interval. For an object traveling in the x-direction, the average velocity may be symbolically represented by

3 average speed vs. average velocity The average speed is the distance traveled by an object divided by the time taken to travel that distance. The magnitude of the average velocity is the change in position of an object divided by the difference in time when the object was located at those positions. Example: A car travels 30 miles east and 10 miles west over a period of 2 hours.

4 Instantaneous velocity The instantaneous velocity at any moment is defined as the average velocity over an infinitesimally short time interval. For an object traveling in the x-direction, the instantaneous velocity may be symbolically represented by

5 Acceleration Acceleration describes how quickly the velocity of an object is changing. The average acceleration is the change in the velocity over some time interval, The instantaneous acceleration at any moment is defined as the average velocity over an infinitesimally short time interval.

6 Motion at constant acceleration Setting, the average velocity is Likewise, the acceleration is, where when the acceleration is constant. Multiplying this last equation by the time, and solve for the velocity gives,

7 Motion at constant acceleration (cont.) Multiplying by the time, and solving for the position gives. Constant acceleration requires the velocity to increase at a uniform rate. Thus, Therefore,

8 Motion at constant acceleration (cont.) Solving the equation,, for time gives Substituting the expression for time into,, gives Therefore,

9 Summary of equations of one-dimensional motion Kinematic equations Constant acceleration time-dependent time-independent

10 1. What is the difference between speed and velocity? A. Speed is an average quantity while velocity is not. B. Velocity contains information about the direction of motion while speed does not. C. Velocity is an average quantity while speed is not. D. The concept of speed applies only to objects that are neither speeding up nor slowing down, while velocity applies to every kind of motion. E. Speed is used to measure how fast an object is moving in a straight line, while velocity is used for objects moving along curved paths.

11 Graphing Motion Motion diagrams Reading values on graph (units) Slopes Areas

12 Representations Motion diagram (student walking to school) Table of data Graph Slide 2-13

13 Making a Motion Diagram

14 Motion Maps Suppose that you took a stroboscopic picture of a car moving to the right at constant velocity where each image revealed the position of the car at equal time intervals. What would it look like if the car were moving faster?

15 The Particle Model A simplifying model in which we treat the object as if all its mass were concentrated at a single point. This model helps us concentrate on the overall motion of the object.

16 Position and Time The position of an object is located along a coordinate system. At each time t, the object is at some particular position. We are free to choose the origin of time (i.e., when t = 0).

17 More complicated motion can be represented as well.

18 Checking Understanding Two runners jog along a track. The positions are shown at 1 s time intervals. Which runner is moving faster?

19 These four motion diagrams show the motion of a particle along the x- axis. Which motion diagrams correspond to a negative acceleration? A. C and D B. B and D C. A, B, C, and D E. A and B D. A and C

20 Graphing constant velocity (position vs. time) a x = 0 for constant velocity. Therefore x = v 0 t + x 0 x (m) 0 t (s) x 0 is not always equal to zero meters.

21 x = v t + x 0 x (m) 0 t (s) In fact, x 0 is not always positive...

22 x = v t + x 0 x (m) t (s) and velocity is not always positive.

23 x = v t + x 0 0 x (m) t (s) Sometimes both position and velocity have negative values.

24 Graphing constant acceleration a (m/s/s) 0 t (s) v (m/s) 0 t (s) The velocity-time graph has a constant slope equal to the acceleration of the object. The acceleration-time graph can be plotted.

25 Write the equation for this line. v (m/s) 0 t (s)

26 v = a t + v 0 v (m/s) 0 t (s) v 0 is the y-intercept or the velocity at time = 0 seconds

27 v = a t + v 0 v (m/s) 0 t (s) The initial velocity may be negative.

28 v = a t + v 0 v (m/s) 0 t (s) The slope of the line for this velocity-time graph is D v/ D t or acceleration ( a )

29 If the velocity is determined by the slope of the position vs. time graph, what will we do with this curved graph? x t

30 Choose 2 points and draw the line. x t

31 Find the slope of that line. x t

32 How does this slope match the velocity of the object? (average velocity over Dt) x x t t t

33 Choose an even smaller interval. How does the slope match the motion? x t x t

34 You probably remember from math class that the best match for the slope of any part of a curve is the slope of the tangent to the curve at that point on the curve. x t

35 What time interval is represented by one point on the curve? x t

36 That point represents the position of the object at an instant of time. x t

37 The slope of the tangent at that point is the velocity of the object at that instant. x t

38 That slope gives the instantaneous velocity. x t

39 The slope of the tangent, instantaneous velocity, changes from one clock reading to the next. x t

40 The slope of the tangent, instantaneous velocity, changes from one clock reading to the next. x t

41 The slope of the tangent, instantaneous velocity, changes from one clock reading to the next. x t

42 The slope of the tangent, instantaneous velocity, changes from one clock reading to the next. x t

43 We can determine the instantaneous velocities at several clock readings... x t

44 Slope of the position-time graph at several clock readings gives velocities at those instants of time. x t Velocities can be plotted on a velocity-time graph. v t

45 Relationships between graphs Slope of the Line Calculates: Velocity Acceleration x v t Area Under The Curve Calculates: Nothing useful to us Displacement Jerk a t t Change in Velocity

46 2. The area under a velocity-versus-time graph of an object is A. the object s speed at that point. B. the object s acceleration at that point. C. the distance traveled by the object. D. the displacement of the object. E. This topic was not covered in this chapter.

47 Here is a motion diagram of a car moving along a straight stretch of road: Which of the following velocity-versus-time graphs matches this motion diagram? A. B. C. D.

48 Velocity increases linearly, so acceleration is a constant. Two methods available Constant acceleration equation Area under the velocity vs. time graph

49 Do A and B ever have the same speed? Do they ever have the same position? 1. Yes 2. No 3. Can t tell!

50 Masses P and Q move with the position graphs shown. Do P and Q ever have the same (instantaneous) velocity? If so, at what time or times? A. P and Q have the same velocity at about 2 s. B. P and Q have the same velocity at 1 s and 3 s. C. P and Q have the same velocity at 1 s, 2 s, and 3 s. D. P and Q never have the same velocity.

51 Determine the slope of the graph shown at point A (time = 2 seconds). What is the area under the graph above between the times of one and three seconds?

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53 A graph of position versus time for a basketball player moving down the court appears like so: Which of the following velocity graphs matches the above position graph? A. B. C. D.

54 A graph of velocity versus time for a hockey puck shot into a goal appears like so: Which of the following position graphs matches the above velocity graph? A. B. C. D.

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56 A cart on a roller-coaster rolls down the track shown below. As the cart rolls beyond the point shown, what happens to its speed and acceleration in the direction of motion? 1. Both decrease. 2. The speed decreases, but the acceleration increases. 3. Both remain constant. 4. The speed increases, but acceleration decreases. 5. Both increase. 6. Other

57 Freefall! Aristotle (b. 384 B.C.) Thought about question of free fall and concluded (albeit incorrectly) that the rate of fall varies and depends on weight. Galileo (b. 1564) Did measurements and found that the rate of fall (acceleration) is constant and independent of weight. Newton (b. 1642) Formulated a coherent theory of motion that became the cornerstone of mechanics and remains so today.

58 Acceleration due to gravity For changes in height near the surface of Earth, we can approximate the acceleration due to gravity as a constant in the downward direction. The magnitude of the acceleration is given by the constant, For freefall, we use the convention of the vertical axis being taken as the y-axis. Assuming upward is the positive direction,. Therefore,

59 A ball is dropped from the roof of a tall building and students in a physics class are asked to sketch a motion diagram for this situation. A student submits the diagram shown. Is the diagram correct? Explain.

60 Challenge Two stones are released from rest at a certain height, one after the other. Will their separation distance 1. increase, 2. decrease, or 3. stay the same?

61 A 1-pound ball and a 100-pound ball are dropped from a height of 10 feet at the same time. In the absence of air resistance A. the 1-pound ball hits the ground first. B. the 100-pound ball hits the ground first. C. the two balls hit the ground at the same time. D. There s not enough information to determine which ball wins the race.

62 Additional Questions Mike jumps out of a tree and lands on a trampoline. The trampoline sags 2 feet before launching Mike back into the air. At the very bottom, where the sag is the greatest, Mike s acceleration is: A. Upward B. Downward C. Zero

63 How to solve problems. (1) Assess what parameters are given and what parameter you are being asked to find. (2) Select the appropriate equation(s) acceleration velocity Known x 0 = 0 m v 0 x f = 275 m v f = 0 a = m/s2 Wanted = 73.4 m/s Math Model:

64 Known x 0 = 0 m a Wanted = 3.06 m/s 2 x f = 2.0 cm v f = 0.35 m/s v 0 = 0 Math Model:

65 A cart rolls down a ramp from rest and travels a distance DX in time Dt. How far will the cart roll from rest in time 2Dt? 1 DX 2 2 DX 3 3 DX 4 4 DX 5 5 DX

66 A cart rolls down a ramp from rest and travels a distance DX in time Dt. How far will the cart roll during the next time interval Dt? 1 DX 2 2 DX 3 3 DX 4 4 DX 5 5 DX

67 The figure below shows five arrows with differing masses that were launched straight up with the noted speeds. Rank the arrows, from greatest to least, on the basis of the maximum height the arrows reach. Ignore air resistance; the only force acting is gravity. 1. E D A B C 2. C D A B E 3. C B A D E 4. E B A D C

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69 Checking Understanding An arrow is launched vertically upward. It moves straight up to a maximum height, then falls to the ground. The trajectory of the arrow is noted. Which graph best represents the vertical velocity of the arrow as a function of time? Ignore air resistance; the only force acting is gravity.

70 If you drop an object in the absence of air resistance, it accelerates downward at 9.80 m/s 2. If instead you throw it downward, its downward acceleration after release is 1. less than 9.80 m/s m/s more than 9.80 m/s 2.

71 An arrow is shot straight up into the air. After 8.0 seconds, the arrow returns to its original height. What was the arrow's initial velocity, ignoring air resistance? (Assume the acceleration of the arrow is g 10 m/s 2.) 1 20 m/s 2 40 m/s 3 60 m/s 4 80 m/s m/s

72 An arrow is shot straight up into the air and it reaches a maximum height of 45 meters. What was the arrow's initial velocity, ignoring air resistance? (Assume the acceleration of the arrow is g 10 m/s 2.) 1 10 m/s 2 20 m/s 3 25 m/s 4 30 m/s 5 35 m/s

73 A person standing at the edge of a cliff throws one ball straight up and another ball straight down at the same initial speed. Neglecting air resistance, the ball to hit the ground below the cliff with the greater speed is the one initially thrown 1. upward. 2. downward. 3. neither they both hit at the same speed.