Representing Motion Chapter 2

Transcription

1 Phenomena

2 Representing Motion Chapter 2

3 Pop Quiz! How fast are you moving at this moment? o A.) 0m/s o B.) 783 mi/h o C.) 350m/s o D.) 30 km/s

4 Pop Quiz! How fast are you moving? oa.) 0m/s ob.) 783 mi/h oc.) 350m/s od.) 30 km/s Speed of the earth s rotation around its 40 o N latitude Speed of the earth s revolution around the sun

5 Pop Quiz! How fast are you moving? oa.) 0m/s ob.) 783 mi/h oc.) 350m/s od.) 30 km/s Speed of the earth s rotation around its 40 o N latitude Speed of the earth s revolution around the sun oin order to describe motion, you need a frame of reference!

6 Frame of Reference A system of objects that are not moving with respect to one another

7 Relative Motion Movement in relation to a frame of reference. Different frames of reference will give the same object different motion.

8 Coordinate Systems

9 Coordinate Systems

10

11 Coordinate Systems

12 Distance The length of a path between two points SI Unit meter Only length of the path is important o Direction doesn t matter o Scalar (magnitude only)

13 Displacement The strait line distance and direction between two points. Vector (magnitude and direction)

14 t = t f t 0 Time Intervals Where: t 0 = initial time t f = final time t = time interval o = change

15 x = x f x 0 Displacement Where: x 0 = initial position: x f = final position x = change in position

16 Displacement If displacements are in the same direction, add the vectors. If displacements are in different directions, subtract the vectors =5 3-2 =1

17 Scalar Vector vs. Scalar Quantities o Magnitude only Vector o Magnitude and Direction

18 Scalar Quantities Vector Quantities Distance Velocity Displacement Time Temperature Force

19 Vectors and Motion A quantity that requires both a magnitude (or size) and a direction can be represented by a vector. Graphically, we represent a vector by an arrow. The velocity of this car is 100 m/s (magnitude) to the left (direction). This boy pushes on his friend with a force of 25 N to the right.

20 Displacement Vectors A displacement vector starts at an object s initial position and ends at its final position. It doesn t matter what the object did in between these two positions. In motion diagrams, the displacement vectors span successive particle positions. Slide 1-29

21 Making a Motion Diagram Slide 1-11

22 Examples of Motion Diagrams Slide 1-12

23 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. Slide 1-13

24 Velocity Vectors Slide 1-34

25 Exercise Alice is sliding along a smooth, icy road on her sled when she suddenly runs headfirst into a large, very soft snowbank that gradually brings her to a halt. Draw a motion diagram for Alice. Show and label all displacement vectors. Slide 1-30

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

27 Checking Understanding Two runners jog along a track. The times at each position are shown. Which runner is moving faster? They are both moving at the same speed. Slide 1-20

28 Speed (m) The car moves 40 m in 1 s. Its speed is = 40 m 1 s. The bike moves 20 m in 1 s. Its speed is = 20 m 1 s. Slide 1-22

29 Velocity Slide 1-23

30 Speed and Velocity Speed vs. Velocity o Speed Scalar o Velocity - Vector Average speed = total distance total time Average velocity: v ҧ = x = x f x 0 t t f t 0 o SI Unit: meter per second (m/s)

31 Position vs. Time Graph Give a qualitative description of the motion of the object depicted in the graph.

32 Position vs. Time Graph The object remained stationary for 1s. It then moved 40m in the positive direction for 2s. It remained stationary for 1.5s and moved 40m in the negative direction for 1.5s.

33 Position vs. Time Graph What is the total distance the object traveled? What is the object s displacement?

34 Position vs. Time Graph What is the object s average velocity between 1s and 3s? v ҧ = x = x f x 0 = 60m 20m t t f t 0 3s 1s = 40m 2s = 20m/s

35 Position vs. Time Graph What is the object s average velocity between 1s and 6s? v ҧ = x = x f x 0 = 20m 20m t t f t 0 6s 1s = 0m 5s = 0m/s What is the object s average speed between 1s and 6s? Average speed = total distance total time = 40m+40m 5s = 80m 5s = 16m/s

36 Position vs. Time Graph The average velocity of an object during the time interval t is equal to the slope of the straight line joining the initial and final points on a graph of the object s position versus time.

37 Position vs. Time Graphs The meaning of Shape Constant Velocity Positive Velocity Changing Velocity Positive Velocity Constant Velocity Slow, Rightward (+) Constant Velocity Fast, Rightward (+) Constant Velocity Slow, Leftward (+) Constant Velocity Fast, Leftward (+) Negative (-) Velocity Slow to Fast Leftward (-) Velocity Fast to Slow

38 Check Your Understanding Use the principle of slope to describe the motion of the objects depicted by the two plots below. In your description, be sure to include such information as the direction of the velocity vector (i.e., positive or negative), whether there is a constant velocity or an acceleration, and whether the object is moving slow, fast, from slow to fast or from fast to slow. Be complete in your description.

39 Position vs. Time Graphs The meaning of Slope The slope of the line on a position versus time graph is equal to the velocity of the object. Slope = y = y 2 y 1 = rise x x 2 x 1 run To determine the slope: Pick two points on the line and determine their coordinates. Determine the difference in y-coordinates of these two points (rise). Determine the difference in x-coordinates for these two points (run). Divide the difference in y-coordinates by the difference in x-coordinates (rise/run or slope). Check Your Understanding: Determine the velocity (i.e., slope) of the object as portrayed by the graph below.

40 x Tangent Lines t SLOPE On a position vs. time graph: VELOCITY SLOPE SPEED Positive Positive Steep Fast Negative Negative Gentle Slow Zero Zero Flat Zero

41 Determining the Area on a v-t Graph For velocity vs. time graphs, the area bounded by the line and the axes represents the distance traveled. The diagram shows three different velocity-time graphs; the shaded regions between the line and the axes represent the distance traveled during the stated time interval. The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 6 seconds. This representation of the distance traveled takes on the shape of a rectangle whose area can be calculated using the appropriate equation. The shaded area is representative of the distance traveled by the object during the time interval from 0 seconds to 4 seconds. This representation of the distance traveled takes on the shape of a triangle whose area can be calculated using the appropriate equation. The shaded area is representative of the distance traveled by the object during the time interval from 2 seconds to 5 seconds. This representation of the distance traveled takes on the shape of a trapezoid whose area can be calculated using the appropriate equation. The method used to find the area under a line on a velocity-time graph depends on whether the section bounded by the line and the axes is a rectangle, a triangle or a trapezoid. Area formulae for each shape are given below.

42 Describing Motion with Velocity vs. Time Graphs - Slope Check Your Understanding The velocity-time graph for a two-stage rocket is shown below. Use the graph and your understanding of slope calculations to determine the acceleration of the rocket during the listed time intervals. o o o a. t = 0-1 second b. t = 1-4 second c. t = 4-12 second

43 Practice Problem Calculate the speed of a dog running through a field if he is covering 23.7 meters in 54 seconds.

44 Practice Problem Which object has a greater velocity, a ball rolling down a 3.4 meter hill in six seconds or a fish swimming upstream and covering 5.4 meters in 0.4 minutes?

45 Practice Problem Calculate the velocity of a mountain climber if that climber is moving northeast at a pace of 1.6 km in 1.4 hours? Give your answer in the SI unit for velocity.

46 Practice Problem If a cyclist in the Tour de France traveled southwest a distance of 12,250 meters in one hour, what would the velocity of the cyclist be?

47 Instantaneous Velocity The speed and direction of an object at a particular moment. The instantaneous velocity v is the limit of the average velocity as the time interval t becomes infinitesimally small x v = lim t 0 t

48 Instantaneous Velocity Positions of a Car at Specific Instants of Time t(s) d(m) Calculated Values of the Time Intervals, Displacements, and Average Velocities for the Car Time Interval (s) t (s) d (m) v ҧ (m/s) 1.00 to to to to to to

49 d (m) Instantaneous Velocity Motion of Car t (s)

50 Equation for Motion for Average Velocity x = vt ҧ + x 0 Look familiar???? y = mx + b Comparison of Straight Lines with Position-Time Graphs General Variable y Motion Variable x m x vҧ t b x 0

51