Velocity Time Graphs 12.2

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1 1. Velocity Time Graphs How are velocities represented on a graph? You can translate the situation shown in Figure 1 into a velocity time graph by first assigning one direction, for example east, as the positive direction, and therefore west as the negative direction. You know that a graph can have positive and negative axes. Therefore, you can show the velocities of both the car and truck on a velocity time graph by using the negative y-axis (below the x-axis) for velocities in the negative direction (Figure ). N W E S 9 km/h 6 km/h Figure 1 A car is travelling at a constant velocity of 9 km/h [E] while a truck is travelling at a constant velocity of 6 km/h [W]. The size of each instantaneous velocity is measured on each vehicle s speedometer. The direction of each is determined from a map or perhaps an onboard compass. Just as the slope of a speed time graph represents acceleration, the slope of a velocity time graph also represents acceleration. Acceleration can be calculated as the change in velocity over time. Now, however, the acceleration has a direction associated with it, and so is a vector quantity. The symbol for vector acceleration is written as a. In Figure, the car and the truck are each travelling at a constant velocity. This is shown by the horizontal lines (zero slope) on the velocity time graph. Now let s consider some changes in velocity shown on velocity time graphs. +9 Velocity (km/h [E]) Time (min) -6 Figure A car is travelling at 9 km/h [E] while a truck is travelling at 6 km/h [W]. 45 Chapter 1

2 Sample Problem 1 A jogger is out for a run. The velocity time graph is shown in Figure 3. For each lettered section of the graph, describe the motion of the jogger. Be sure to specify the direction of the motion. 3 A A Jogger Goes for a Run Velocity (m/s [N]) B D C Time (h) E F Figure 3 A jogger goes for a long run. According to the velocity-time graph, the jogger A. runs north at a constant velocity of.5 m/s for about the first two hours; B. slows and then stops at the end of. h; C. remains stationary for about.5 h; D. accelerates rapidly toward the south after about.5 h; E. runs at. m/s [S] from about.5 h to about 3.5 h; F. abruptly changes her velocity to 1. m/s [S] (a walk) for about.5 h before stopping. Constant Acceleration Now let s consider an example of uniformly-changing velocity (constant acceleration). For example, suppose a bird, such as a kingfisher, dives from rest and drops under the influence of gravity. If we assume there is no friction, the downward velocity will increase uniformly. rise slope = r u n (from a velocity time graph) a = v Remember that we are now considering acceleration to be a vector quantity. In Chapter 1 we calculated the size of the acceleration without considering the direction. Now we will communicate both the size and the direction of the acceleration. If the object s velocity is increasing, the direction of the acceleration is always the same as the direction of the velocity. If the velocity is established as north and the velocity increases to the north, then the acceleration is north. If the velocity is north but decreasing, then the acceleration is south. Fortunately, using these conventions, we can easily find the direction of the acceleration from the calculation. Even if the motion is in the up/down dimension, one of these directions can be assigned a positive sign. Displacement, Velocity, and Acceleration 453

3 Sample Problem What is the acceleration of the diving kingfisher in Figure 4? Up is the positive direction and down is negative. v 1 =. m/s v = 8. m/s t 1 =. s t =.8 s rise slope = r u n a = v = v v 1 ( 8. (.)) m/s = (.8.) s = 1 m/s According to the slope of the velocity-time graph, the average acceleration of the kingfisher is 1 m/s [down]. Velocity (m/s [up]) Time (s) v A Diving Kingfisher Figure 4 The slope of a velocity-time graph represents the kingfisher s acceleration. Average Velocity As you may recall from Chapter 1, the average speed of an object in motion can be determined from the ratio of total distance divided by total elapsed time. Similarly, the average velocity of an object in motion can be determined from the ratio of total (resultant) displacement divided by total time. v av = d R Sample Problem 3 In January of Canadian Jeremy Wotherspoon (Figure 5) broke his own speed skating world record in the 5-m event with a time of s. (a) What was his average speed during the skate? (b) What was the average velocity of his skate? Assume that north is positive and that the start and finish line were in the same place. (a) v av = d (b) v av = d R 5 m = = m [ N] s s = 14.3 m/s = m/s [N] Figure 5 Jeremy Wotherspoon may have been disappointed by his average velocity. 454 Chapter 1

4 Jeremy Wotherspoon skated at an average speed of 14.3 m/s during his world-record-breaking skate. However, his average velocity is zero since his resultant displacement was zero. He started and finished the race at the same position. The average velocity of a uniformly accelerating object can also be determined in the same way as the average of most things: adding the largest value and the smallest value and then dividing by two: v av = v 1 + v The average velocity has the same value as the instantaneous velocity at the mid-time point for constant acceleration (Figure 6). It is important to note that the formula given above and the velocity at the mid-time point only work when the velocity-time graph is straight (indicating uniform acceleration). Finding Average Velocity from a Graph v Velocity v av v 1 t Time Figure 6 To find the average velocity, locate the mid-time point, and read the velocity at that time. Sample Problem 4 What is the average velocity during the interval of a car crash if the initial velocity is 1 m/s [NE] and the car comes to rest.11 s later? (Assume that the car crash involves constant acceleration.) Let the north-east direction be positive. v 1 = +1 m/s v = m/s v av = v 1 + v (+1 m/s) + ( m/s) = = +6. m/s The average velocity during the interval of the car crash is 6. m/s [NE]. Displacement, Velocity, and Acceleration 455

Velocity (m/s [E]) Understanding Concepts 1. What information does the slope of a velocity time graph give us?. A tennis player practises by hitting a tennis ball toward a brick wall.

What is the average velocity in the following situations? (a) You walk to school and back home. (b) A car accelerates uniformly from rest to 6 km/h [W].

5 Velocity (m/s [E]) Understanding Concepts 1. What information does the slope of a velocity time graph give us?. A tennis player practises by hitting a tennis ball toward a brick wall. Interpret the velocity time graph in Figure 7 to describe what is happening to the ball in each segment of its trip Velocity of a Tennis Ball Time (s) 4. What is the average velocity in the following situations? (a) You walk to school and back home. (b) A car accelerates uniformly from rest to 6 km/h [W]. (c) The instantaneous velocity at the midpoint of constant positive acceleration is 45 km/h [S]. (d) A diver enters the water at 17 m/s [down] and is stopped by the water within.5 m. What assumption is made in this calculation? 5. In a -m race, an athlete accelerates rapidly out of the blocks for 3 s before the false start pistol is fired. The athlete stops within s, and then takes 1 s to walk slowly back to the starting blocks. Sketch a velocity time graph for this athlete. Compare this graph to the position time graph that you drew in question 1 in Video analysis of a space shuttle launch (Figure 9) provides a table of evidence (Table 1). Figure 7 3. The Queen s Plate is a horse race for Canadian-bred threeyear-old horses. The race is run at Woodbine Racetrack in Toronto, Ontario (Figure 8). The record for the 1-m race is : (two minutes and two seconds). The horses start running westward, go past the finish, complete a circuit of the track, and finish in front of the stands, 7 m [W] of where they began. (a) What is the average speed for the record run? (b) What is the average velocity for the record run? Figure 8 Figure 9 Table 1 Launching a Space Shuttle Interval Total time (s) Velocity (m/s [up]) (a) Draw a velocity time graph from this evidence and V determine the acceleration of the shuttle during this segment of the launch. (The best-fit line need not pass through the origin.) 456 Chapter 1 SKILLS HANDBOOK: V Graphing

6 (b) Does the acceleration of the space shuttle appear constant? Explain. (c) If the acceleration is constant, use your graph to determine the average velocity of the space shuttle over the entire 5. s. 7. A car, leaving a city speed zone of 45 km/h [E], accelerates uniformly to the new speed limit of 15 km/h [E] in 7. s. What is the average velocity of the car during this constant acceleration? 8. Refer to Figure 4 on page 454. (a) If downward is defined as the positive direction, how would the graph change? (b) Recalculate Sample Problem (page 454) using down as positive. (c) Which method do you prefer? Why? 9. Draw a velocity time graph and a speed time graph for a shopping trip in a car from home that involves (a) accelerating quickly to 4 km/h [S]; (b) driving at a constant 4 km/h [S] for 1 min; (c) stopping to shop for 1 min; (d) accelerating rapidly to 4 km/h [N]; (e) driving at 4 km/h [N] for 1 min; (f) stopping at home. 1. How can you determine the instantaneous velocity from (a) a straight line on a position time graph? (b) a curved line on a position time graph? (c) a straight line on a velocity-time graph? (d) a curved line on a velocity-time graph? 1. When the police investigate accident scenes, they collect a variety of evidence. What are some examples of information that might be recorded? What kinds of results might the police hope to obtain from the analysis of the evidence? Exploring 13. What information can we find from (a) the area under the curve of a velocity time graph? (b) the area under the curve of a position time graph? 14. Look again at Figure (page 45). (a) List at least four pieces of information that you can obtain from the graph. (b) What information can you not obtain from the graph? 15. Draw a velocity vector diagram to represent the shopping trip in question Which, of the two graphs in question 9 and the vector diagram in question 15, provides the best information? Explain. Reflecting 17. In the two sections of this chapter that you have just completed, you have learned about position time and velocity time graphs. How did understanding distance time and speed time graphs help you to understand these new graphs? What is the main difference between these new graphs and the old ones? Making Connections 11. Driving habits in city traffic have a variety of environmental and economic consequences. Sketch a velocity-time graph for a poor or undesirable driving habit in stop-and-go traffic, and another graph illustrating a better or more desirable driving habit for the same trip. Be prepared to discuss the differences in the two graphs from at least two perspectives. Displacement, Velocity, and Acceleration 457

Calculating Acceleration

Calculating Acceleration Textbook pages 392 405 Before You Read Section 9. 2 Summary How do you think a velocity-time graph might differ from the position-time graph you learned about in the previous chapter?