SPEED AND VELOCITY LEARNING GOALS Students will: Know the definitions of average speed and average velocity. Solve motion problems that use the concepts of average speed and average velocity. PREPARATION AT HOME Reading Nelson Physics 11 Section 1.2 Pages 14-20 Physics Classroom Speed and Velocity Videos Khan Academy o Calculating Average Velocity or Speed o Displacement from Time and Velocity o Solving for Time Reading Quiz AVERAGE SPEED When travelling a distance in a car, you are travelling at a certain speed. A speedometer tells you how many kilometers you can travel over the time interval of one hour. The average speed of a moving object is the total distance covered per unit time. The SI unit for speed is metres per second (m/s). Like distance, speed is a scalar quantity. Given what you know about total distance travelled and a time interval ( ), derive an equation to find the average speed ( ) using words and variables. EXAMPLE 1: CALCULATING AVERAGE SPEED Your dog runs in a straight line for a distance of 43 m in 28 s. What is your dog s average speed? AVERAGE VELOCITY In addition to knowing how slow or fast an object moves, it is also important to know the direction of a moving object. In this case, you will need to use the displacement, which is a vector quantity. Using the displacement will allow you to find the average velocity of a moving object. The average velocity ( ) is found by calculating the total displacement over the total time interval for that displacement to be accomplished. Express the equation for average velocity in terms of words and variables. Compare what you wrote with how it is written on your formula sheet. 1
Position (m) SPH3U1 Lesson 02 Kinematics A position-time graph is a graph that describes the motion of an object, with position on the vertical axis and time on the horizontal axis. Whenever an object is moving at a constant velocity, the position-time graph of that motion is a straight line. The slope of a position-time graph gives the velocity of the object. The steeper the graph, the greater is the object s displacement in a given time interval, and the higher is its velocity. EXAMPLE 2: SOLVING FOR THE VELOCITY WITH A POSITION-TIME GRAPH The following table shows the position of a golf ball as it moves away from you in 1.0 s intervals. The position-time graph for the golf ball is also shown. Calculated the average velocity (slope) of the golf ball. Time (s) Position (m) 0.0 0.0 1.0 1.0 2.0 2.0 3.0 3.0 4.0 4.0 5.0 5.0 5.0 4.0 3.0 2.0 1.0 Position vs Time of a Golf Ball 0.0 0.0 1.0 2.0 3.0 4.0 5.0 Time (s) EXAMPLE 3: SOLVING PROBLEMS USING THE EQUATION FOR AVERAGE VELOCITY Find the average velocity of a student who jogs 750 m [E] in 5.0 min, stops and does static stretches for 10.0 min, and then walks another 3.0 km [E] in 30.0 min. Note that on a position-time graph, the rise is which is the total displacement and the run is which is the total elapsed time. To find the average velocity in the above problem, you found the total displacement (3.75 km [E]) and divided by the total time (5.0 min +10.0 min + 30.0 min = 45.0 min = 0.75 h) POSITIVE AND NEGATIVE AVERAGE VELOCITY Using position-time graphs we can find the average velocity by finding the slope. In example 2, you found the slope of a graph which gave you a positive velocity. Based on the displacement of an object, when will an object obtain a positive average velocity? 2
Similarly, when will an object obtain a negative average velocity? ZERO AVERAGE VELOCITY There are two ways in which an object can obtain a zero average velocity. The average velocity is determined by the displacement. If an object has not moved, its initial and final position will be the same and have an average velocity of zero. An object s initial and final position could also be the same even if the object has moved. Consider a train starting at Union Station in Toronto. The train travels for five hours to a station in Ottawa. Over this part of the trip, the train has a positive average velocity. The same train makes a return trip to Toronto and ends at Union Station. The train has returned to its initial position! Over the second part of the trip, the train has a negative average velocity. Over the full return trip, the initial and final positions are the same and the displacement is zero, which gives a zero average velocity. Develop another situation with your group in which an object would obtain a zero average velocity and yet have travelled a distance. UNIFORM MOTION (CONSTANT VELOCITY) AND NON-UNIFORM VELOCITY Uniform motion or constant velocity is motion at a constant speed in a straight line. It is the simplest type of motion that an object can undergo, except for being at rest. In contrast, motion with non-uniform velocity is motion that is not at a constant speed or not in a straight line. Motion with non-uniform velocity may also be called accelerated motion. INSTANTANEOUS VELOCITY The moment-to-moment measure of an object s velocity is called its instantaneous velocity. A vehicle speedometer tells you the instantaneous velocity. Example 2 shows a case of constant velocity (uniform motion). The graph shows a constant unchanging slope. At every point, the instantaneous velocity is equal to the instantaneous velocity at every other point. This is also equal to the overall average velocity. EXAMPLE 4 A quarterback is trying to avoid being tackled. He runs 10.0 m [left] in 2.3 s. He then runs 25.0 m [right] in 4.4 s. What is his average speed and velocity? Redo the above problem incorrectly find the average speed by finding the average for each section and then add the two results and divide by 2. Discuss why this is wrong. 3
PRACTICE PROBLEMS 1. A camper kayaks 16 km [E] from a camping site, stops, and paddles 23 km [W]. a. What is the camper's final position with respect to the campsite? b. What is the total displacement of the camper? c. What is the distance covered by the camper? Is it the same as the displacement? Explain. 2. Two SCUBA divers take turns riding an underwater tricycle at an average speed of 1.74 km/h for 60.0 h. What distance do they travel in this time? 3. An airplane is travelling from Vancouver to Toronto following the jet stream. The plane cruised at an average speed of 1100 km/h for the 4000 km flight. How long did the flight take? If the plane left Vancouver at 3:00 am EST, what time did it arrive in Toronto? 4. A truck driver, reacting quickly to an emergency, applies the brakes. During the driver s 0.32 s reaction time, the truck maintains a constant velocity of 27 m/s [fwd]. What is the displacement of the truck during the time the driver takes to react? 5. A swimmer crosses a circular pool with a radius of 16 m in 21 s. a. What is the swimmer s average speed? b. If the swimmer were to swim around the circumference of the pool at this same speed, how long would it take? 6. A city bus leaves the terminal and travels, with a few stops, along a straight route that takes it 12 km [E] of it starting position in 24 minutes. In another 18 minutes, the bus turns around and retraces its path, ending at a stop 2.0 km [E] of the terminal. What is the average speed of the bus for the entire route? 7. The same bus as in question 6 is on the same route. Determine a. its average velocity from the terminal to the farthest position from the terminal. b. its average velocity for the entire trip. c. Explain why your answers for a and b are different. 8. A truck travels at an average speed of 45 km/h over a distance of 105 km. It then travels another 85 km at a higher average speed. His overall average speed for the entire trip is 55 km/h. What was his average speed for the second stage? 9. The Arctic tern holds the world record for bird migration distance. The tern migrates once a year from islands north of the Arctic Circle to the shores of Antarctica, a displacement of approximately 1.6 x 10 4 km [S]. (The route, astonishingly, lies mainly over water.) If a tern s average velocity during this trip is 21 km/h [S], how long does the journey take? (Convert your answer to days). 10. Bugs Bunny travels from his rabbit hole for 25 minutes to the farmer s field 3.5 km [E] of his hole. However, when he arrives, there is the farmer waiting with a gun so Bugs scoots back towards his hole and hides under a bush that is 1.5 km [E] of his hole. His dash took 5.0 minutes. a. Determine Bugs average speed. b. Determine Bugs average velocity. 4
11. For the graph given: a. Describe the motion of the object in stages A, B, and C. b. What is the displacement of the object during each stage? c. What is the total displacement (between the initial and final positions)? d. What is the total distance covered? 12. For each line on the graph given: a. What is the displacement in each case? b. Is it necessary that the position points for an object and its final displacement have the same sign? Explain using the graph. 13. A wildlife biologist measures how long it takes four animals to cover a displacement of 200 m [forward]. Graph the data below, how long does it take the Elk and Grizzly bear to cover 150 m? Animal Time Taken (s) Elk 10.0 Coyote 10.4 Grizzly Bear 18.0 Moose 12.9 14. Two sprinters are racing. Billy has a head start of 3.0 s and is travelling at 8 m/s. Biff travels at 10 m/s. If the race is 200 m, who would win? By how much? Answers: 1. a) 7 km [W] of campsite b) 7 km [W] c) 39 km 2. 104 km 3. 3.64 h; 6:38 am 4. 8.64 m [fwd] 5. a) 1.5 m/s b) 66 s 6. 31.4 km/h 7. a) 30 km/h [E] b) 2.9 km/h [E] 8. 75.8 km/h 9. 31.7 d 10. a) 11 km/h b) 3 km/h [E] 11. b) +16 m, 0 m, -16 m c) 0 m d) 32 m 12. A) -10 m B) -25 m C) +25 m 14. Billy 25 s; Biff 20 s + 3 s = 23 s; Biff wins by 2 s 5