POSITION, MOTION AND DISPLACEMENT LEARNING GOALS Students will: Define what is meant by a vector quantity and by a scalar quantity. Understand the concept of position (a vector quantity). Relate a change in position to the concept of displacement. Understand that displacements can be added. MOTION Mechanics - The field of physics in which we study the motion of objects. Kinematics The study of motion without referring to its cause. SCALARS AND VECTORS In order to locate objects, you need a reference system. The first step is to choose a reference point, also called the origin. Based on the placement of the origin, you can indicate an object s position and distance from the origin. Consider the hockey player to the right. The reference point chosen is center dot. You can then measure the straight-line distance,, of the player from the origin. If you specify a direction along with the distance, then you define the player s position,. The player s position would also include its direction relative to the reference point. The direction is indicated after the units and enclosed in square brackets (ex. 6.0 m [north]). A. What is the initial position ( of the hockey player shown in the diagram (assume a NSEW coordinate system with the nets at the W and E ends of the rink)? Figure 1 The player's position and distance are measured relative to the reference point. The arrow over the variable indicates it is a vector quantity. A vector has both magnitude (number value) and direction, while a scalar has only a magnitude. If the player moves towards the east end of the rink, their position and distance have changed. B. What is the player s final position ( ) and straight-line distance ( ) travelled from his initial position? Figure 2 The player's position has changed which results the player having a displacement. The distance travelled is the length of the path taken to move from one position to another, regardless of direction (i.e. it does not have to be a straight-line path). The displacement,, is the change in position, where is the Greek letter delta and represents change in. Delta is defined mathematically as final value minus initial value. In other words, displacement has to be a straight-line path with the direction of that path specified. 1
C. How might you find the player s displacement based on the positions you found earlier? Develop an equation for finding an object s displacement from its initial and final values. D. What does it mean when an object has experienced a displacement in a given direction? SIGN CONVENTIONS An object may also move in multiple directions. In order to discover an object s displacement in this situation, consider your origin to be a part of a coordinate axis with a positive and a negative direction. E. Draw a coordinate system indicating both positive and negative directions, a system indicating the four points of a compass and a system indicating up, down, left and right. F. Develop a scenario where an object begins at a positive position and travels to a negative position. Include the magnitudes and directions of your object. An object s position can be written in two separate ways; with a positive or negative sign in front of the number value or with a direction included after the units. G. Write the object s initial and final positions using both methods. H. In your scenario above, what is the object s displacement? 2
EXAMPLE 1 A dog has an initial position 3.4 m [W] of his dog house. He then walks to the other side so he is 1.6 m [E] of his dog house. What was his displacement? Do this problem with both an equation and with a diagram. Sometimes, you are not given positions; instead, you are given more than one displacement and asked to find the total displacement. EXAMPLE 2 A basketball player fakes out the defence by stepping 0.75 m [right] and then moving 3.50 m [left]. What is the player s total displacement? VECTOR SCALE DIAGRAMS In the previous example, you used algebra to solve a displacement problem. You can also use vector diagrams to solve problems. A vector can be represented by a directed line segment, which is a straight line between two points with a specific direction. A directed line segment representing a vector always has two ends. The line segment has an arrowhead at one end referred to as the tip or head. The other end is the tail. The initial point is represented by the tail end and the final point as the tip. Draw a line with an arrowhead at one end that connects the points A to B. Label the initial point tail and final point tip. A B EXAMPLE 3 A traveller initially standing at a bus stop moves 1.5 m to the right and then moves 5.0 m to the left. Determine the traveller s total displacement (a) using directions and (b) using plus and minus signs. Draw a vector scale diagram below to help you. (Hint: Draw two vectors representing the movement that are added by placing them tip to tail. Tip to tail means the second vector is started where the first vector ended) Scale: 1.0 cm = 1.0 m 3
PRACTICE PROBLEMS 1. Copy the image below to answer the following questions algebraically and using vector scale diagrams. MALL H SCHOOL LIBRARY 500 m west 1000 m 1200 m east a. Imagine that you walk from home to school in a straight-line route. What is your displacement? b. What is your displacement if you walk from your school to the library? c. One night after working at the library, you decide to go to the mall. What is your total displacement when walking from the library to the mall? 2. A dog is practising for her agility competition. She leaves her trainer and runs 80 m due west to pick up a ball. She then carries the ball 27 m due east and drops it into a bucket. What is the dog s total displacement? 3. A golfer hits a ball from a golf tee at a position of 16.4 m [W] relative to the clubhouse. The ball comes to rest at a position of 64.9 m [W] relative to the clubhouse. Determine the displacement of the golf ball. 4. A rabbit runs 3.8 m [N] and stops to nibble on some grass. The rabbit then hops 6.3 m[n] to scratch against a small tree. What is the rabbit s total displacement. 5. A car drives 73 m [W] to a stop sign. It then continues on for a displacement of 46 m [W]. Use a vector scale diagram to determine the car s total displacement. 6. A robin flies 32 m [S] to catch a worm and then flies 59 m [N] back to its nest. Use a vector scale diagram to determine the robin s total displacement. Answers: 1. a) 500 m [E] b) 700 m [E] c) 2200 m [W] 2. 53 m W 3. 48.5 m [W] 4. 10.1 m [N] 5. 120 m [W] 6. 27 m [N] 4
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. 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. A. 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. B. Express the equation for average velocity in terms of words and variables. 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. 5
Position (m) SPH3U1 Lesson 01 Kinematics 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. C. Based on the displacement of an object, when will an object obtain a positive average velocity? D. Similarly, when will an object obtain a negative average velocity? 6
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. E. Develop another situation 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? F. 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. 7
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. 8
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 9