Kinematics Motion in 1-Dimension

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1 Kinematics Motion in 1-Dimension Lana Sheridan De Anza College Jan 15, 219

2 Last time how to solve problems 1-D kinematics

3 Overview 1-D kinematics quantities of motion graphs of kinematic quantities vs time

4 Position Quantities position displacement distance #» x #» x = #» x f #» x i d Position and displacement are vector quantities. Position and displacement can be positive or negative numbers. Distance is a scalar. It is always a positive number. Units: meters, m

5 point. A graphical representation of thi 3 Such a plot is called a position time graph Notice the alternative representations o 2 38 motion The starting position of the car is #» of the car. Figure 2.1a is a pictor 3 graphical x i = 3 representation. m î, the final Table position 2.1 is a tabu #» x f = 5 m4 î. 237 Using an alternative representation is oft the situation in a given problem. The u The distance the car travels is d = 5 m 3 m = 2 m. Position, Displacement, Distance Example the right between #» The displacement of the car is x #» x = #» x f #» x i = 2 m î

6 Position, Displacement, Distance Example The starting position of the car is #» x i = 3 m î, the final position is #» x f = 6 m î. The distance the car travels is d = 13 m. the right between x The displacement of the car is a the left between x positions = #» x f and #» x i. = ( 6 î) 3 î m = 9 m î x b

7 Speed We need a measure how fast objects move. speed = distance time If an object goes 1 m in 1 second, its speed is 1 m/s.

8 Speed Speed can change with time. For example, driving. Sometimes you are on the highway going fast, sometimes you wait at a stoplight. Instantaneous speed is an object s speed at any given moment in time.

9 Speed Speed can change with time. For example, driving. Sometimes you are on the highway going fast, sometimes you wait at a stoplight. Instantaneous speed is an object s speed at any given moment in time. Average speed is the average of the object s speed over a period of time: average speed = total distance traveled time interval

10 Velocity Driving East at 65 mph is not the same as driving West at 65 mph.

11 Velocity Driving East at 65 mph is not the same as driving West at 65 mph. There is a quantity that combines the speed and the direction of motion. This is the velocity.

12 Velocity Driving East at 65 mph is not the same as driving West at 65 mph. There is a quantity that combines the speed and the direction of motion. This is the velocity. Velocity is a vector quantity. Speed is a scalar quantity.

13 Velocity Driving East at 65 mph is not the same as driving West at 65 mph. There is a quantity that combines the speed and the direction of motion. This is the velocity. Velocity is a vector quantity. Speed is a scalar quantity. If a car drives in a circle, without speeding up or slowing down, is its speed constant?

14 Velocity Driving East at 65 mph is not the same as driving West at 65 mph. There is a quantity that combines the speed and the direction of motion. This is the velocity. Velocity is a vector quantity. Speed is a scalar quantity. If a car drives in a circle, without speeding up or slowing down, is its speed constant? Is its velocity constant?

15 Velocity How position changes with time. Quantities velocity average velocity instantaneous speed average speed #» v v #» x avg = #» t v or #» v (= d #» x dt ) d t

16 Velocity How position changes with time. Quantities velocity average velocity instantaneous speed average speed #» v v #» x avg = #» t v or #» v (= d #» x dt ) d t Can velocity be negative?

17 Velocity How position changes with time. Quantities velocity average velocity instantaneous speed average speed #» v v #» x avg = #» t v or #» v (= d #» x dt ) d t Can velocity be negative? Can speed be negative?

18 Velocity How position changes with time. Quantities velocity average velocity instantaneous speed average speed #» v v #» x avg = #» t v or #» v (= d #» x dt ) d t Can velocity be negative? Can speed be negative? Units: meters per second, m/s

19 point. A graphical representation of thi 3 Such a plot is called a position time graph The displacement of the car is x #» Notice the alternative representations o 2 38 motion = 2of mthe î. car. Figure 2.1a is a pictor 3 The distance the car travels is d graphical = 2 m. representation. Table 2.1 is a tabu Using an alternative representation is oft The time for 5 the car to 253 move this the far situation is 1 seconds. in a given problem. The u What is the average velocity of the car? What is the average speed of the car? Average Velocity vs Average Speed Example the right between #» x

20 point. A graphical representation of thi 3 Such a plot is called a position time graph The displacement of the car is x #» Notice the alternative representations o 2 38 motion = 2of mthe î. car. Figure 2.1a is a pictor 3 The distance the car travels is d graphical = 2 m. representation. Table 2.1 is a tabu Using an alternative representation is oft The time for 5 the car to 253 move this the far situation is 1 seconds. in a given problem. The u What is the average velocity of the car? What is the average speed of the car? Average Velocity vs Average Speed Example the right between average velocity #» x v avg = #» t = 2 m/s î (same 1 2 in this 3 case) average speed = d t = 2 m/s #» x

21 5 253 Average Velocity vs Average Speed Example The displacement of the car is 9 m î. The distance the car travels is d = 13 m. The time for the car to move A F is 5 seconds. Average velocity? Average speed? the situation in a given problem. The u the right between x x the left between 4 6 a b

22 5 253 Average Velocity vs Average Speed Example The displacement of the car is 9 m î. The distance the car travels is d = 13 m. The time for the car to move A F is 5 seconds. Average velocity? Average speed? the situation in a given problem. The u the right between x x a the xleft between t average velocity #» v avg = #» = 1.8 m/s î average speed = d t = 2.6 m/s Not the same! 4 6 b

23 Question Quick Quiz Under which of the following conditions is the magnitude of the average velocity of a particle moving in one dimension smaller than the average speed over some time interval? A A particle moves in the +x direction without reversing. B A particle moves in the x direction without reversing. C A particle moves in the +x direction and then reverses the direction of its motion. D There are no conditions for which this is true. 1 Serway & Jewett, page 24.

24 Question Quick Quiz Under which of the following conditions is the magnitude of the average velocity of a particle moving in one dimension smaller than the average speed over some time interval? A A particle moves in the +x direction without reversing. B A particle moves in the x direction without reversing. C A particle moves in the +x direction and then reverses the direction of its motion. D There are no conditions for which this is true. 1 Serway & Jewett, page 24.

25 Conceptual Question y rankings, remember that zero is greater than a negative value. If two values are equal, show that they are equal in your ranking.) c Conceptual Questions 1. denotes answer available in Student So 1. If the average velocity of an object is zero in some time interval, what can you say about the displacement of the object for that interval? 2. Try the following experiment away from traffic where you can do it safely. With the car you are driving moving slowly on a straight, level road, shift the transmission into neutral and let the car coast. At the moment the car comes to a complete stop, step hard on the brake and notice what you feel. Now repeat the same experiment on a fairly gentle, uphill slope. Explain the difference in what a person riding in the car feels in the two cases. (Brian Popp suggested the idea for this 1 Serway & Jewett, page Y g w a w e 7. ( b i z 8. ( b

26 Acceleration Speed and velocity can change with time. Acceleration is the rate of change of velocity with time. acceleration = change in velocity time interval

27 Acceleration Speed and velocity can change with time. Acceleration is the rate of change of velocity with time. acceleration = change in velocity time interval If an object is moving with constant speed in a circular path, is it accelerating?

28 Acceleration Quantities acceleration average acceleration #» a a #» v avg = #» t Acceleration is also a vector quantity.

29 Acceleration Quantities acceleration average acceleration #» a a #» v avg = #» t Acceleration is also a vector quantity. If the acceleration vector is pointed in the same direction as the velocity vector (ie. both are positive or both negative), the particle s speed is increasing.

30 Acceleration Quantities acceleration average acceleration #» a a #» v avg = #» t Acceleration is also a vector quantity. If the acceleration vector is pointed in the same direction as the velocity vector (ie. both are positive or both negative), the particle s speed is increasing. If the acceleration vector is pointed in the opposite direction as the velocity vector (ie. one is positive the other is negative), the particle s speed is decreasing. (It is decelerating.)

31 Graphing Kinematic Quantities One very convenient way of representing motion is with graphs that show the variation of these kinematic quantities with time. Time is written along the horizontal axis we are representing time passing with a direction in space (the horizontal direction).

32 Table 2.1 Position of the Car at Various Times Position t (s) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 5 m to the left of x 5 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between t b x t (s) Figures from Serway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

33 Table 2.1 Position of the Car at Various Times Position t (s) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 5 m to the left of x 5 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between t b x t (s) Figures from Serway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

34 Table 2.1 Position of the Car at Various Times Position t (s) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 5 m to the left of x 5 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between t b x t (s) Figures from Serway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

35 Table 2.1 Position of the Car at Various Times Position t (s) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 5 m to the left of x 5 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between t b x t (s) Figures from Serway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

36 Table 2.1 Position of the Car at Various Times Position t (s) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 5 m to the left of x 5 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between t b x t (s) Figures from Serway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

37 Table 2.1 Position of the Car at Various Times Position t (s) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 5 m to the left of x 5 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between t b x t (s) Figures from Serway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

38 Table 2.1 Position of the Car at Various Times Position t (s) vs. Time Graphs origin of coordinates (see Fig. 2.1a). It continues moving to the left and is more than 5 m to the left of x 5 when we stop recording information after our sixth data point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between t b x t (s) Figures from Serway & Jewett Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. (b) A graphical representation (position time graph) of the motion of the car.

39 Position t (s) 3 Position vs. Time Graphs point. A graphical representation of this information is presented in Figure 2.1b. Such a plot is called a position time graph. Notice the alternative representations of information that we have used for the motion of the car. Figure 2.1a is a pictorial representation, whereas Figure 2.1b is a graphical representation. Table 2.1 is a tabular representation of the same information. Using an alternative representation is often an excellent strategy for understanding the situation in a given problem. The ultimate goal in many problems is a math- the right between a the left between b t x t (s) The average velocity in the interval A B is the slope of the blue line connecting the points A and B. v avg = x Figure 2.1 A car moves back and forth along a straight line. Because we are interested only in the car s translational motion, we can model it as a particle. Several representations of the information about the motion of the car can be used. Table 2.1 is a tabular representation of the information. (a) A pictorial representation of the motion of the car. t (b) A graphical representation (position time graph) of the motion of the car. 1 Figures from Serway & Jewett

40 Average Velocity in Position vs. Time Graphs Chapter 2 Motion in One Dimension A B: v avg = x t = 2 m/s i A F: v a avg = x t = 1.8 m/s i 3 4 t (s) Figure 2.3 (a) Graph representing the motion of the car in F b

41 Velocity in Position vs. Time Graphs The (instantaneous) velocity is the rate of change of displacement the slope of a velocity-time graph. apter 2 Motion in One Dimension y graph of represents in the n the vertiin the quan- a Pos.-time graph for car t (s) zoomed in represents the velocity of the car at point. What we have done is determine the instantaneous velocity at that moment. t In other t words, the instantaneous velocity v x equals the limiting value of the ratio Dx/Dt as Dt approaches zero: 1 b The blue line between positions and approaches the green tangent line as point is moved closer to point. The green line is the tangent line, gives the slope of the curve at t =. x v = lim Figure 2.3 (a) Graph representing the motion of the car in Figure 2.1. (b) An enlargement of the upper-left-hand corner of the graph.

42 Summary quantities of motion graphs of position vs time Homework Walker Physics: prev: Ch 2, onward from page 47. Conc. Ques: 1, 3, 9; Probs: 1, 3, 5, 7, 9, 13 new: Ch 2, onward from page 47. Conc. Ques: 13; Probs: 35

Kinematics Motion in 1-Dimension

Kinematics Motion in 1-Dimension Lana Sheridan De Anza College Jan 16, 2018 Last time unit conversions (non-si units) order of magnitude calculations how to solve problems Overview 1-D kinematics quantities