Comparing Graphs of Linear Motion PDF

Comparing Graphs of 1.4 Linear Motion Cheetahs are adapted for speed—they are the fastest land animals. They can accel- erate at faster rates than most sports cars (Figure 1). Cheetahs have been measured accel...

Comparing Graphs of 1.4 Linear Motion Cheetahs are adapted for speed—they are the fastest land animals. They can accel- erate at faster rates than most sports cars (Figure 1). Cheetahs have been measured accelerating at rates greater than 10 m/s2. To put this in perspective, a sports car can accelerate at approximately 7.2 m/s2. In fact, cheetahs are capable of accelerating from rest to 10 m/s in only three strides. You have already seen how position–time and velocity–time graphs can be used to analyze the linear motion of objects. In this section, we will introduce acceleration– time graphs and use all three types of graphs to analyze motion in more detail. Figure 1 Cheetahs have the greatest acceleration of any animal. Acceleration–Time Graphs Earlier in this chapter, you learned how to find the displacement, or change in posi- tion, of an object by determining the area under a velocity–time graph. In a similar way, we can determine the change in velocity of an object from the area under an acceleration–time graph, which has acceleration on the vertical axis and time on the acceleration–time graph a graph horizontal axis. describing motion of an object, with Consider the acceleration–time graph in Figure 2, which shows the motion of a acceleration on the vertical axis and cheetah. The points plotted on this graph lie along a horizontal straight line with a time on the horizontal axis non-zero y-intercept. The acceleration is a constant 4.0 m/s2, so this graph represents uniform acceleration. Acceleration v. Time for Motion with Uniform Acceleration 5.0 4.0 a (m/s2 [W]) 3.0 2.0 1.0 0 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 t (s) Figure 2 Acceleration–time graph showing motion with uniform acceleration NEL 1.4 Comparing Graphs of Linear Motion   31  Investigation 1.4.1 If we calculate the area under the acceleration–time graph in Figure 2 from 0 s to 5.0 s, we will be determining the change in velocity of the object from t 5 0 s to Uniform Velocity (p. 47) t 5 5.0 s: In this investigation, you will use a A 5 lw motion sensor to generate different types of motion graphs for an object m 5 15.0 s2 a4.0 3W 4 b moving with uniform velocity, and s2 analyze these graphs. A 5 20 m/s 3 W 4 Since the units are metres per second, the area we calculated represents a change in velocity. The area under an acceleration–time graph represents the change in velocity of an object. If the initial velocity of the cheetah is zero (the object is at rest), the final velocity is equal to the change in velocity, 20 m/s [W]. If the initial velocity is 5 m/s [W], however, then the graph tells us that the final velocity is 5 m/s [W] 1 20 m/s [W] 5 25 m/s [W] The graph does not tell us what the initial and final velocities are; it just tells us the change in velocity that occurs in the time interval. Relationships among Linear Motion Graphs Investigation 1.4.2 Graphical analysis is one of the most powerful analytical tools available to physi- cists. In studying the motion of objects, analyzing position–time, velocity–time, and Motion Down a Ramp (p. 48) acceleration–time graphs can help us gain insight into real-life events such as the In this investigation, you will use a motion of the cheetah shown in Figure 1. This is particularly important because most motion sensor and different types of objects in nature do not come equipped with a speedometer. motion graphs to analyze the motion Figure 3 compares the three types of graphs of linear motion. All three graphs of an object rolling down a ramp. represent the same type of motion: uniform acceleration. Nevertheless, the three graphs look very different. When analyzing a motion graph, you may read informa- tion directly from the graph or determine further information by calculating the slope or the area of the graph. a (m/s2 ) area area v (m/s) d (m) slope slope t (s) t (s) t (s) Figure 3 Position–time, velocity–time, and acceleration–time graphs of the same motion 32   Chapter 1 Motion in a Straight Line NEL  Tutorial 1 Creating One Type of Motion Graph from Another By using the information in Figure 4, we can analyze an acceleration–time graph further and get more information about the motion it describes. Sample Problem 1: Creating a Velocity–Time Graph from an Acceleration–Time Graph Use the acceleration–time graph in Figure 4 to generate velocity Step 3. P lot the data to create a velocity–time graph and time data for the object. Then use these data to plot a (Figure 5). velocity–time graph. Step 1. To generate the velocity–time data, first calculate Velocity v. Time for Motion with Uniform Acceleration the area under the graph for several time points in Figure 4. 25.0 Since the line is horizontal, we use the formula for the area 20.0 v (m/s) [W]) of a rectangle, A 5 lw. > 15.0 For Figure 4, l 5 t 1s2 ; w 5 a 1m/s2 3 W 42 ; and > 10.0 A 5 v 1m/s 3 W 42 , so in calculating A = lw, we are > > 5.0 actually calculating v 5 1Da 2 1Dt 2 0 Acceleration v. Time for Motion with 0 1.0 2.0 3.0 4.0 5.0 Uniform Acceleration t (s) 5.0 Figure 5 Velocity–time solution graph 4.0 a (m/s2 [W]) 3.0 Figure 5 shows the resulting graph. It is an increasing straight 2.0 line with a zero intercept. It describes precisely the same motion 1.0 that was described by the acceleration–time graph in Figure 4. 0 Both graphs describe uniform acceleration. 0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 t (s) Figure 4 Using an acceleration–time graph to create other motion graphs Step 2. Table 1 shows the calculations for the area under the graph at 1 s intervalsOntario from t = Physics 0 s to t =11 5.0Us. 0176504338 Table 1 Calculating the Velocity at Various Time Points in Figure 4 FN C01-F30-OP11USB Time Acceleration Equation Velocity > CO> > > CrowleArt Group t (s) a 1m/s2 3 W 42 v 5 1Da 2 1Dt 2 v 1m/s2 3 W 4 Deborah Crowle 0 4.0 > Pass m 3rd pass0 v 5 a4.0 2 3 W 4 b 10 s2 s Approved 1.0 4.0 > Not Approved m 4.0 11 U v 5 a4.0 2 3 W 4 b 11.0 s2 s 2.0 C01-F29-OP11USB 4.0 > m 8.0 v 5 a4.0 2 3 W 4 b 12.0 s2 CrowleArt Group s Deborah Crowle3.0 4.0 > m 12 v 5 a4.0 2 3 W 4 b 13.0 s2 3rd pass s 4.0 4.0 > m 16 v 5 a4.0 2 3 W 4 b 14.0 s2 s 5.0 4.0 > m 20.0 v 5 a4.0 2 3 W 4 b 15.0 s2 s NEL 1.4 Comparing Graphs of Linear Motion    33  Sample Problem 2: Creating an Acceleration–Time Graph from a Velocity–Time Graph Use the velocity–time graph shown in Figure 6 to plot the rise slope 5 corresponding acceleration–time graph. run > > Dv Velocity v. Time for Motion with a av 5 Dt Uniform Acceleration 14.0 7.8 m/s 3 W 4 5 13.0 3.0 s > 12.0 a av 5 2.6 m/s2 [W] 11.0 Step 2. Use this value to create an acceleration–time graph 10.0 (Figure 7). 9.0 Acceleration v. Time for Motion with 8.0 Uniform Acceleration v (m/s [W]) 3.0 a (m/s2 [W]) 7.0 2.0 6.0 1.0 5.0 0 4.0 0 1.0 2.0 3.0 4.0 5.0 3.0 t (s) 2.0 Figure 7 Acceleration–time solution graph 1.0 Figure 7 shows the corresponding acceleration–time graph. 0 This graph shows a horizontal straight line with a y-intercept of 0 1.0 2.0 3.0 4.0 5.0 2.6 m/s2 [W]. t (s) Note: If a velocity–time graph is not a straight line, you will need to Figure 6 Given velocity–time graph determine the slope of the tangent for each time data point, and then use these data to plot the corresponding acceleration–time graph. Step 1. The data plotted on the velocity–time graph in Figure 6 form an increasing straight-line graph with a zero intercept. You can determine acceleration from a velocity–time graph by calculating its slope. Since the velocity–time graph in Figure 6 is a straight line, its slope does not change. OntarioSo Physics we can calculate 11 U the slope or acceleration over any 0176504338 time interval. FN C01-F32-OP11USB Practice CO CrowleArt Group Velocity v. Time for Motion with 1. Generate position–time and acceleration–time data Deborah Crowle Uniform Acceleration representing the motion of the object shown in Figure 8. 8.0 Pass 2nd pass Use the data to plot the corresponding position–time and Approved 7.0 acceleration–time graphs. T/I C 31-OP11USB Not Approved 6.0 leArt Group 5.0 v (m/s) [N] rah Crowle 4.0 pass 3.0 2.0 1.0 0 0 0.5 1.0 1.5 2.0 t (s) Figure 8 34   Chapter 1 Motion in a Straight Line NEL  1.4 Summary The area under an acceleration–time graph gives the velocity of the object. Given one type of motion graph, you can read or calculate data from it in order to construct a different type of graph. 1.4 Questions 1. Copy and complete Table 2 in your notebook by adding a check mark in each column that applies. K/U Table 2 How do you Read information determine … Given a … from graph Take the slope Find the area position position–time graph velocity position–time graph velocity velocity–time graph velocity acceleration–time graph acceleration velocity–time graph acceleration acceleration–time graph 2. From the velocity–time graph in Figure 9, generate Position v. Time for Accelerated Motion position–time data and then plot the corresponding 70.0 position–time graph, assuming the initial position is 60.0 0 m. T/I C 50.0 d (m [S]) 40.0 Velocity v. Time for Complex Motion 10.0 30.0 8.0 20.0 v (m/s) [S] 6.0 10.0 4.0 0 0 1.0 2.0 3.0 4.0 5.0 6.0 2.0    Figure 10 t (s) 0 0 1.0 2.0 3.0 4.0 5.0 6.0 t (s) 4. Use the data in the velocity–time graph shown in Figure 11    Figure 9 to plot the corresponding acceleration–time graph. T/I C 3. Consider the position–time graph shown in Figure 10. T/I Velocity v. Time for Accelerated Motion (a) What is the position of the object at t 5 5.0 s? t (s) (b) What is the instantaneous velocity of the object at 0 1.0 2.0 3.0 4.0 5.0 6.0 t 5 3.0 s? 0 v (m/s [S]) (c) What is the average velocity for the object’s motion –5.0 from 0 s to 6.0 s? –10.0 –15.0    Figure 11 Ontario Physics 11 U 0176504338 NEL FN C01-F35-OP11USB 1.4 Comparing Graphs of Linear Motion    35 CO CrowleArt Group U