Tactics Box 2.1 Interpreting Position-versus-Time Graphs

Transcription

1 1D kineatic Retake Assignent Due: 4:32p on Friday, October 31, 2014 You will receive no credit for ites you coplete after the assignent is due. Grading Policy Tactics Box 2.1 Interpreting Position-versus-Tie Graphs Learning Goal: To practice Tactics Box 2.1 Interpreting position-versus-tie graphs. A continuous curve that shows an object's position as a function of tie is called a position-versus-tie graph. You can gain relevant inforation on an object's otion if you interpret such graphs correctly. To do that, it is useful to review the definition of velocity. For otion along a line, the velocity v x is the ratio of the displaceent Δx of an object to the tie interval Δt during which this displaceent occurs, which can be written as v x v x This equation has a graphical interpretation: It tells us that is the slope of the position-versus-tie graph representing the otion. This iplies that you can associate the slope of the graph, a geoetrical quantity, with the physical quantity velocity. This and other aspects of interpreting position-versus-tie graphs are outlined in the following Tactics Box. Δx Δt. TACTICS BOX 2.1: Interpreting position-versus-tie graphs Inforation about otion can be obtained fro position-versus-tie graphs as follows: 1. Deterine an object's position at tie t by reading the graph at that instant of tie. 2. Deterine the object's velocity at tie t by finding the slope of the position graph at that point. The agnitude of the velocity is defined as speed. Steeper slopes correspond to faster speeds. 3. Deterine the direction of otion by noting the sign of the slope. Positive slopes correspond to positive velocities and, hence, to otion to the right (or up). Negative slopes correspond to negative velocities and, hence, to otion to the left (or down). The picture on the left shows the position-versus-tie graph of an object that is oving along a straight line. Find the object's positions x1, x2, x3, and x4 at ties t1 2.0 s, t2 4.0 s, t3 13 s, and t4 17 s. Express your answers in eters to one significant figures, separated by coas. 1/33

2 ,,, x1 x2 x3 x4,,, Part B Find the object's speeds,, and at ties t1, t2, and t3. v1 v2 v3 2.0 s 4.0 s 13 s Express your answers in eters per second to two significant figures, separated by coas.,, v1 v2 v3 /s, /s, /s Part C At any oent in tie, the object can be stationary, oving to the right, oving to the left, or turning around. Identify the otion of the object at each tie provided. Conceptual Question /33

3 Choose the correct velocity-versus-tie graph for a rock that is thrown straight upward, fro the instant it leaves the hand until the instant it hits the ground. 3/33

4 A Flea in Flight In this proble, you will apply kineatic equations to a juping flea. Take the agnitude of free-fall acceleration to be Ignore air resistance. /s 2 A flea jups straight up to a axiu height of What is its initial velocity v0 as it leaves the ground? Express your answer in eters per second to three significant figures. Hint 1. Finding the knowns and unknowns Take the positive y direction to be upward, the y coordinate of the initial position of the flea to be y0, and denote the final height of the flea by y1, whose value you know. Let t be the duration of the flea's leap to its axiu height, v0 its initial velocity, v1 its final velocity (at axiu height), and ay its (constant) acceleration. Which of the following quantities is/are known? Check all that apply. 0 Hint 1. The nuber of known quantities Typically, you need to know the values of four variables in order to solve any of the kineatic equations, because they contain five variables each, with the exception of v1 v0 + ayt, which contains only four variables, in which case you would need to know the values of only three of these variables. Since we ay place the flea at any point of the y axis to begin its jup, we have conveniently assued that y 0 is equal to 0. Hint 2. What is the flea's velocity at its axiu height? What is the velocity v1 of the flea at its axiu height of y ? Express your answer in eters per second to three significant figures. v1 /s 4/33

5 t v0 v1 ay Hint 2. Deterine which kineatic equation to use Decide which kineatic equation akes the solution of this proble easiest. That is, look for an equation that contains the variable you are solving for and in which all the other variables are known. v1 v0 + ayt y 1 y 0 + v 0 t+ 1 2 ayt 2 v 2 1 v ay( y 1 y 0 ) v0 v1 ( )t + y 1 y 0 2 Hint 3. Soe algebra help You have deterined that the siplest equation to use is v 2 1 v ay( y 1 y 0 ). To solve for v0, you ust first subtract the ter 2 ay( y 1 y 0 ) fro both sides of the equation, and then take the square root of both sides. Keep in ind that the acceleration is negative. v0 /s Part B How long is the flea in the air fro the tie it jups to the tie it hits the ground? Express your answer in seconds to three significant figures. Hint 1. How to approach the proble One approach is to find the tie it takes for the flea to go fro the ground to its axiu height, and then find the tie it takes for the flea to fall fro its axiu height to the ground. The subsequent hints will guide you through this approach. Hint 2. Find the tie fro the ground to the flea's axiu height 5/33

6 What is the tie tup it takes the flea to go fro the ground ( y 0 0, v0 ) to its axiu height ( y , v1 )? Express your answer in seconds to three significant figures. tup s Hint 3. Find the tie fro the flea's axiu height to the ground What is the tie tdown that it takes for the flea to fall fro its axiu height ( y , v0 0 ) to 0 the ground ( y1 )? Express your answer in seconds to three significant figures. tdown s tie in air s Proble 2.18 The figure shows a graph of actual position-versus-tie data for a particular type of drag racer known as a "funny car." s Estiate the car's velocity at /33

7 Express your answer to two significant figures and include the appropriate units. v x Part B Estiate the car's velocity at 4.0. s Express your answer to two significant figures and include the appropriate units. v x Proble 2.26 When striking, the pike, a predatory fish, can accelerate fro rest to a speed of 3.5 in /s s What is the acceleration of the pike during this strike? Express your answer using two significant figures. a /s 2 Part B How far does the pike ove during this strike? Express your answer using two significant figures. d Proble 2.45 A gannet is a seabird that fishes by diving fro a great height. 7/33

8 /s If a gannet hits the water at 32, what height did it dive fro? Assue that the gannet was otionless before starting its dive. Express your answer to two significant figures and include the appropriate units. h Proble 2.48 A football is kicked straight up into the air; it hits the ground 5.0 later. s What was the greatest height reached by the ball? Assue it is kicked fro ground level. Express your answer to two significant figures and include the appropriate units. y ax Part B With what speed did it leave the kicker's foot? Express your answer to two significant figures and include the appropriate units. v y What x vs. t Graphs Can Tell You To describe the otion of a particle along a straight line, it is often convenient to draw a graph representing the position of the particle at different ties. This type of graph is usually referred to as an x vs. t graph. To draw such a graph, choose an axis syste in which tie t is plotted on the horizontal axis and position x on the vertical axis. Then, indicate the values of x at various ties t. Matheatically, this corresponds to plotting the variable x as a function of t. An exaple of a graph of position as a function of tie for a particle traveling along a straight line is shown below. Note that an x vs. t graph like this does not represent the path of the particle in space. Now let's study the graph shown in the figure in ore detail. Refer to this graph to answer Parts A, B, and C. 8/33

9 What is the total distance Δx traveled by the particle? Express your answer in eters. Hint 1. Total distance The total distance Δx traveled by the particle is given by the difference between the initial position x0 at and the position at. In sybols, t 0.0 s x t 50.0 s Δx x x0. Hint 2. How to read an x vs. t graph Reeber that in an x vs. t graph, tie t is plotted on the horizontal axis and position x on the vertical axis. For exaple, in the plot shown in the figure, at. x 16.0 t 10.0 s Δx Part B What is the average velocity vav of the particle over the tie interval? Express your answer in eters per second. Δt 50.0 s Hint 1. Definition and graphical interpretation of average velocity The average velocity vav of a particle that travels a distance Δx along a straight line in a tie interval Δt is defined as 9/33

10 vav In an x vs. t graph, then, the average velocity equals the slope of the line connecting the initial and final positions. Δx Δt. Hint 2. Slope of a line ( ta, xa ) (, ) The slope of a line fro point A, with coordinates, to point B, with coordinates tb xb, is equal to the "rise" over the "run," or x B tb xa. ta vav /s Part C What is the instantaneous velocity v of the particle at? Express your answer in eters per second. t 10.0 s Hint 1. Graphical interpretation of instantaneous velocity The velocity of a particle at any given instant of tie or at any point in its path is called instantaneous velocity. In an x vs. t graph of the particle's otion, you can deterine the instantaneous velocity of the particle at any point in the curve. The instantaneous velocity at any point is equal to the slope of the line tangent to the curve at that point. v /s Another coon graphical representation of otion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of tie. In this graph, tie t is plotted on the horizontal axis and velocity v on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line otion, however, these vectors have only one nonzero coponent in the direction of otion. Thus, in this proble, we will call v the velocity and a the acceleration, even though they are really the coponents of the velocity and acceleration vectors in the direction of otion. Part D Which of the graphs shown is the correct v vs. t plot for the otion described in the previous parts? 10/33

11 Hint 1. How to approach the proble Recall your results found in the previous parts, naely the fact that the instantaneous velocity of the particle is constant. Which graph represents a variable that always has the sae constant value at any tie? Graph A Graph B Graph C Graph D Part E Shown in the figure is the v vs. t curve selected in the previous part. What is the area A of the shaded region under the curve? Express your answer in eters. 11/33

12 Hint 1. How to approach the proble The shaded region under the v vs. t curve is a rectangle whose horizontal and vertical sides lie on the t axis and the v axis, respectively. Since the area of a rectangle is the product of its sides, in this case the area of the shaded region is the product of a certain quantity expressed in seconds and another quantity expressed in eters per second. The area itself, then, will be in eters. A Velocity fro Graphs of Position versus Tie An object oves along the x axis during four separate trials. Graphs of position versus tie for each trial are shown in the figure. 12/33

13 During which trial or trials is the object's velocity not constant? Check all that apply. You did not open hints for this part. Trial A Trial B Trial C Trial D Part B During which trial or trials is the agnitude of the average velocity the largest? Check all that apply. You did not open hints for this part. Trial A Trial B Trial C Trial D Proble 2.4 A rural ail carrier is driving slowly, putting ail in ailboxes near the road. He overshoots one ailbox, stops, shifts into reverse, and then backs up until he is at the right spot. The velocity graph of the figure represents his otion. 13/33

14 Draw the ail carrier's position-versus-tie graph. Assue that at. x 0 t 0 s Part B What is the position of the ailbox? Express your answer to two significant figures and include the appropriate units. 14/33

15 x What Velocity vs. Tie Graphs Can Tell You A coon graphical representation of otion along a straight line is the v vs. t graph, that is, the graph of (instantaneous) velocity as a function of tie. In this graph, tie t is plotted on the horizontal axis and velocity v on the vertical axis. Note that by definition, velocity and acceleration are vector quantities. In straight-line otion, however, these vectors have only a single nonzero coponent in the direction of otion. Thus, in this proble, we will call v the velocity and a the acceleration, even though they are really the coponents of the velocity and acceleration vectors in the direction of otion, respectively. Here is a plot of velocity versus tie for a particle that travels along a straight line with a varying velocity. Refer to this plot to answer the following questions. What is the initial velocity of the particle, v0? Express your answer in eters per second. You did not open hints for this part. v0 /s Part B What is the total distance Δx traveled by the particle? Express your answer in eters. 15/33

16 You did not open hints for this part. Δx Part C What is the average acceleration aav of the particle over the first 20.0 seconds? Express your answer in eters per second per second. You did not open hints for this part. aav /s 2 Part D What is the instantaneous acceleration a of the particle at? t 45.0 s You did not open hints for this part. a 1 /s 2 /s 2 /s 2 /s 2 /s Now that you have reviewed how to plot variables as a function of tie, you can use the sae technique and draw an acceleration vs. tie graph, that is, the graph of (instantaneous) acceleration as a function of tie. As usual in these types of graphs, tie t is plotted on the horizontal axis, while the vertical axis is used to indicate acceleration a. Part E Which of the graphs shown below is the correct acceleration vs. tie plot for the otion described in the previous parts? 16/33

17 You did not open hints for this part. Graph A Graph B Graph C Graph D Proble 2.20 x t s We set the origin of a coordinate syste so that the position of a train is 0 at 0. shows the train's velocity graph. 17/33

18 Choose the correct position graph for the train. 18/33

19 Part B Choose the correct acceleration graph for the train. 19/33

20 Part C Find the acceleration of the train at t 3.0. s Express your answer to two significant figures and include the appropriate units. 20/33

21 a x PSS 2.1: Motion with constant acceleration Learning Goal: To practice Proble-Solving Strategy 2.1 Motion with constant acceleration You are driving down the highway late one night at 20 when a deer steps onto the road 35 in front of you. Your reaction tie before stepping on the brakes is 0.50 s, and the axiu deceleration of your car is 10 /s 2. How uch distance is between you and the deer when you coe to a stop? PROBLEM-SOLVING STRATEGY: Motion with constant acceleration Probles involving constant acceleration speeding up, slowing down, vertical otion, horizontal otion can all be treated with the sae proble-solving strategy. PREPARE: Draw a visual overview of the proble. This should include a otion diagra, a pictorial representation, and a list of values; a graphical representation ay be useful for certain probles. SOLVE: The atheatical solution is based on the three equations /s ( v x ) f ( v x ) i + a x Δt x f x i +( v x ) Δt + 1 i 2 a x (Δt) 2 ( ( + 2 Δx v x ) 2 f v x ) 2 i a x Though the equations are phrased in ters of the variable x, it is custoary to use y for otion in the vertical direction. Use the equation that best atches what you know and what you need to find. For exaple, if you know acceleration and tie and are looking for a change in velocity, the first equation is the best one to use. Unifor otion with constant velocity has zero acceleration. ASSESS: Is your result believable? Does it have proper units? Does it ake sense?,. (a 0), Prepare Get a sense of how the situation works by graphing the velocity. Then deterine which variables are known and which are unknown. Which of the following shows the shape of the velocity versus tie graph for the situation described above? 21/33

22 A B C D Part B 0 s 0.5 s Sort the variables for the period fro t1 to t2 based on whether their values are known or unknown. Hint 1. Identifying the initial and final quantities In this part, you are asked about the tie period fro t1 to t2. Therefore, xi and vi correspond to x1 and v1. Siilarly, xf and vf correspond to x2 and v2. Copare the figure given after to the inforation in the introduction to deterine which of these quantities are given and which need to be found. 22/33

23 Part C Sort the variables for the period fro t2 until the car coes to rest, at tie t3, based on whether their values are known or unknown. 0.5 s Hint 1. Identifying the initial and final quantities In this part, you are asked about the tie period fro t2 to t3. Therefore, xi and vi correspond to x2 and v2. Siilarly, xf and vf correspond to x3 and v3. Copare the figure given after to the inforation in the introduction to deterine which of these quantities are given and which need to be found. 23/33

24 Solve Use kineatic equations to find your final position and then copare it to the position of the deer. Part D How uch distance is between you and the deer when you coe to a stop? Express your answer in eters. Hint 1. How to approach the proble To find the distance between you and the deer, you need to know the position of the deer and your position when you coe to a stop. Use the proper kineatic equations to find the final position for the first tie interval discussed in the Prepare step. Then, use a kineatic equation and the final position fro the first interval to find the final position for the second tie interval (i.e., the position when you stop). Hint 2. Find the deer's position Using the coordinate syste with x1 position? Express your answer in eters. 0 and the deer in the positive x direction, what is the deer's x Hint 3. Find your position when you stop Using the coordinate syste with x1 when you stop? 0 and the deer in the positive x direction, what is your position 24/33

25 Express your answer in eters. Hint 1. Find your position when you press the brakes The first of the two tie intervals that you identified in the Prepare step is fro t1 to t2 0.5 s. The final position x2 for this interval is your position when you press the brakes. What is this final position? Express your answer in eters. 0 s Hint 1. Select the proper kineatic equation Which of the three kineatic equations given in the proble-solving strategy should you use to find the position when you press the brakes? Recall that in Part B, you found that all of the variables for this tie interval are known except for xf, which is what you are looking for. ( ( + Δt v x ) f v x ) i a x x f xi +( v x ) Δt + 1 i 2 a x (Δt) 2 ( v x ) 2 f ( v x ) 2 i + 2a x Δx xf Hint 2. Find the distance traveled in the second tie interval The second tie interval is fro t2 until the car stops at tie t3. Find the distance Δx traveled in the second tie interval. Express your answer in eters. 0.5 s Hint 1. Select the proper kineatic equation Which of the kineatic equations fro the proble-solving strategy should you use to deterine the distance traveled in the second tie interval? ( ( + Δt v x ) f v x ) i a x +( Δt + (Δt x f x i v x ) i 1 2 a x ) 2 ( ( + 2 Δx v x ) 2 f v x ) 2 i a x 25/33

26 Δx x Assess Deterine whether your answer akes sense. Part E You found the distance that you will cover while braking. You can use the expected aount of tie for braking to check your answer for distance. Using the inforation given in the introduction and the value for distance that you found above, deterine the tie required for you to stop once you press the brakes. Note that the tie for braking is just the tie interval t3 t2. Express your answer in seconds. t3 t2 s Proble 2.40 When a jet lands on an aircraft carrier, a hook on the tail of the plane grabs a wire that quickly brings the plane to a halt before it overshoots the deck. In a typical landing, a jet touching down at 240 is stopped in a distance of 95. k/h What is the agnitude of the jet's acceleration as it is brought to rest? Express your answer to two significant figures and include the appropriate units. a x 26/33

27 Part B How uch tie does the landing take? Express your answer to two significant figures and include the appropriate units. Δt Proble 2.38 A light-rail train going fro one station to the next on a straight section of track accelerates fro rest at 1.1 for 20 s /s 2. It then proceeds at constant speed for 1100 before slowing down at 2.2 until it stops at the station. /s 2 What is the distance between the stations? Express your answer using two significant figures. d Part B How uch tie does it take the train to go between the stations? Express your answer using two significant figures. t s Tossing Balls off a Cliff Learning Goal: To clarify the distinction between speed and velocity, and to review qualitatively one-diensional kineatics. A woan stands at the edge of a cliff, holding one ball in each hand. At tie t0, she throws one ball straight up with speed v0 and the other straight down, also with speed v0. For the following questions neglect air resistance. Pay particular attention to whether the answer involves "absolute" quantities that have only agnitude (e.g., speed) or quantities that can have either sign (e.g., velocity). Take upward to be the positive direction. 27/33

28 If the ball that is thrown downward has an acceleration of agnitude a at the instant of its release (i.e., when there is no longer any force on the ball due to the woan's hand), what is the relationship between a and g, the agnitude of the acceleration of gravity? a > g a g a < g Part B Which ball has the greater acceleration at the instant of release? the ball thrown upward the ball thrown downward Neither; the accelerations of both balls are the sae. Part C Which ball has the greater speed at the instant of release? You did not open hints for this part. the ball thrown upward the ball thrown downward Neither; the speeds are the sae. Part D Which ball has the greater average speed during the 1-s interval after release (assuing neither hits the ground during that tie)? You did not open hints for this part. 28/33

29 the ball thrown upward the ball thrown downward Neither; the average speeds of both balls are the sae. Part E Which ball hits the ground with greater speed? the ball thrown upward the ball thrown downward Neither; the balls hit the ground with the sae speed. Conceptual Question 2.2 Choose an exaple of a vertical otion with a positive velocity and a negative acceleration. An elevator start fro the ground floor and oves to the top floor of a building. A ball dropped fro a height during its downward otion. Parachutist descending at a constant speed with an open parachute. A ball thrown upward, before it starts to fall back down. Part B Choose an exaple of a vertical otion with a negative velocity and a negative acceleration. A ball thrown upward, before it starts to fall back down. Parachutist with an open parachute descending at a constant speed. An elevator starts fro the ground floor and oves to the top floor of a building. A ball dropped fro a height during its downward otion. Multiple Choice Question /33

30 The figure shows the position graph of a car traveling on a straight road. The velocity at instant 1 is and the velocity at instant 2 is. positive, negative positive, positive negative, negative negative, zero Proble 2.55 The figure shows the otion diagra, ade at two fraes of fil per second, of a ball rolling along a track. The track has a 3.0--long sticky section. 30/33

31 t 0 s What is the change in the ball's position fro to? Express your answer using two significant figures. t 1.0 s Δs Part B t 2.0 s What is the change in the ball's position fro to? Express your answer using two significant figures. t 4.0 s Δs Part C What is the ball's velocity before reaching the sticky section? Express your answer using two significant figures. v s /s Part D What is the ball's velocity after passing the sticky section? Express your answer using two significant figures. v s /s Part E Deterine the ball's acceleration on the sticky section of the track. Express your answer using two significant figures. 31/33

32 a /s 2 Multiple Choice Question 2.17 The figure shows an object's position-versus-tie graph. What is the velocity of the object at 3? /s /s /s /s /s s Multiple Choice Question 2.19 A car is traveling at 20. The driver applies the brakes and the car decelerates at What is the stopping distance? v x /s a x /s /33

33 Score Suary: Your score on this assignent is 0.0%. You received 0 out of a possible total of 21 points. 33/33