4.1 Position, Speed, and Velocity Where are you right now? How fast are you moving? To answer these questions precisely, you need to use the concepts of position, speed, and velocity. These ideas apply to ordinary objects, such as cars, bicycles, and people. They also apply to microscopic objects the size of atoms and to enormous objects like planets and stars. Let s begin our discussion of motion with the concept of position. The position variable Position as a variable You may do an experiment in your class that uses a car on a track. How do you tell someone exactly where the car is at any given moment? The answer is by measuring its position. Position is a variable. The position of the car describes where the car is relative to the track. In the diagram below, the position of the car is 50 centimeters (cm). That means the center of the car is at the 50 cm mark on the track. Position and distance Position and distance are similar but not the same. Both use units of length. However, position is given relative to an origin. The origin is the place where position equals 0 (near the left end of the track above). Here s an example of the difference between position and distance. Assume the track is 1 meter long. Suppose the car moves a distance of 20 cm away from the 50 cm mark. Where is it now? You know a distance (20 cm) but you still don t know where the car is. It could have moved 20 cm to the right or 20 cm to the left. Saying the car is at a position of 70 cm tells you where the car is. A position is a unique location relative to an origin (Figure 4.1). 78 UNIT 2 MOTION AND FORCE position - a variable that tells location relative to an origin. origin - a place where the position has been given a value of zero. Figure 4.1: If the car moves 20 cm to the right, its position will be 70 cm.
Speed Speed is a motion variable The variable speed describes how quickly something moves. To calculate the speed of a moving object, you divide the distance it moves by the time it takes to move. For example, if you drive 120 miles (the distance) and it takes you 2 hours (the time) your speed is 60 miles per hour (60 mph = 120 miles 2 hours). The lower case v letter is used to represent speed. MOTION Chapter 4 speed - describes how quickly an object moves, calculated by dividing the distance traveled by the time it takes. average speed - the total distance divided by the total time for a trip. instantaneous speed - the actual speed of a moving object at any moment. Units for speed The units for speed are distance units over time units. If distance is in kilometers and time in hours, then speed is in kilometers per hour (km/h). Other metric units for speed are cm per second (cm/s) and meters per second (m/s). Your family s car probably shows speed in miles per hour (mph). Table4.1 shows different units commonly used for speed. Distance Time Speed Abbreviation meters seconds meters per second m/s kilometers hours kilometers per hour km/h centimeters seconds centimeters per second cm/s miles hours miles per hour mph Average speed and constant speed When you divide the total distance of a trip by the time taken, you get the average speed. Figure4.2 shows an average speed of 100 km/h. But, think about actually driving though Chicago. On a real trip, your car will slow down and speed up. Sometimes your speed will be higher than 100 km/h, and sometimes lower (even 0 km/h!) The speedometer shows you the car s instantaneous speed. The instantaneous speed is actual the speed an object has at any moment. Figure 4.2: A driving trip with an average speed of 100 km/h. 4.1 POSITION, SPEED, AND VELOCITY 79 Table 4.1: Common Units for Speed
Solving Problems: Speed How far will you go if you drive for 2 hours at a speed of 100 km/h? 1. Looking for: You are asked for a distance. 2. Given: You are given the speed and the time. 3. Relationships: distance = speed time 4. Solution: distance = (100 km/h) (2 h) = 200 km Your turn... a. You travel at an average speed of 20 km/h in a straight line to get to your grandmother s house. It takes you 3 hours to get to her house. How far away is her house from where you started? b. What is the speed of a snake that moves 20 meters in 5 seconds? c. A train is moving at a speed of 50 km/h. How many hours will it take the train to travel 600 kilometers? 80 UNIT 2 MOTION AND FORCE The Speed Limit of the Universe The fastest speed in the universe is the speed of light. Light moves at 300 million meters per second (3 x 108 m/s). If you could make light travel in a circle, it would go around the Earth 7.5 times in one second! Scientists believe the speed of light is the ultimate speed limit in the universe. a. Your grandmother s house is 60km away from where you started. b. The snake s speed is 4 m/s. c. It will take the train 12 hours to travel 600 kilometers.
MOTION Chapter 4 Vectors and velocity Telling in front from behind Using positive and negative numbers How can you tell the difference between one meter in front of you and one meter behind you? The variable of distance is not the answer. The distance between two points can only be positive (or zero). You can t have a negative distance. For example, the distances between the ants in Figure 4.3 are either positive or zero. Likewise, one meter in front of you and one meter behind you both have the same distance: 1 meter. The answer is to use position and allow positive and negative numbers. In the diagram below, positive numbers describe positions to the right (in front) of the origin. Negative numbers are to the left (or behind) the origin. vector - a variable that gives direction information included in its value. velocity - a variable that tells you both speed and direction. Vectors Position is an example of a kind of variable called a vector. A vector is a variable that tells you a direction as well as an amount. Positive and negative numbers are enough information for a variable when the only directions are forward and backward. When up down and right left are also possible directions, vectors get more complicated. Figure 4.3: Distance is always a positive value or zero. The difference between velocity and speed Velocity Like position, motion can go right, left, forward or backward. We use the term velocity to mean speed with direction. Velocity is positive when moving to the right, or forward. Velocity is negative when moving to the left, or backward (Figure 4.4). Velocity is a vector, speed is not. In regular conversation you might use the two words to mean the same thing. In science, they are related but different. Speed can have only a positive value (or zero) that tells you how far you move per unit of time (like meters per second). Velocity is speed and direction. If the motion is in a straight line, the direction can be shown with a positive or negative sign. The sign tells whether you are going forward or backward and the quantity (speed) tells you how quickly you are moving. Figure 4.4: Velocity can be a positive or a negative value. 4.1 POSITION, SPEED, AND VELOCITY 81
Keeping track of where you are A robot uses vectors Pathfinder is a small robot sent to explore Mars (Figure 4.5). As it moved, Pathfinder needed to keep track of its position. How did Pathfinder know where it was? It used its velocity vector and a clock to calculate every move it made. Use two variables to find the third one Any formula that involves speed can also be used for velocity. For example, you move 2 meters if your speed is 0.2 m/s and you keep going for 10 seconds. But did you move forward or backward? You move 2 meters (backwards) if you move with a velocity of 0.2 m/s for 10 seconds. Using the formulas with velocity gives you position instead of distance. Forward and backward movement Suppose Pathfinder moves forward at 0.2 m/s for 10 seconds. Its velocity is +0.2 m/s. In 10 seconds, its position changes by +2 meters. Adding up a series of movements Now, suppose Pathfinder goes backward at 0.2 m/s for 4 seconds. This time the velocity is 0.2 m/s. The change in position is 0.8 meters. A change in position is velocity time (Figure 4.6). The computer in Pathfinder adds up +2 m and 0.8 m to get +1.2 m. After these two moves, Pathfinder s position is 1.2 meters in front of where it was. Pathfinder knows where it is by keeping track of each move it makes. It adds up each change in position using positive and negative numbers to come up with a final position (Figure 4.7). 82 UNIT 2 MOTION AND FORCE Figure 4.5: Pathfinder is a robot explorer which landed on Mars in 1997 (NASA/JPL). Figure 4.6: The change in position or distance is the velocity multiplied by the time. Figure 4.7: Each change in position is added up using positive and negative numbers.
MOTION Chapter 4 Maps and coordinates Two dimensions If Pathfinder was crawling on a straight board, it would have only two choices for direction. Positive is forward and negative is backward. Out on the surface of Mars, Pathfinder has more choices. It can turn and go sideways! The possible directions include north, east, south, west, and anything in between. A flat surface is an example of two dimensions. We say two because it takes two number lines to describe every point (Figure 4.8). North, south, east, and west Coordinates describe position One way to describe two dimensions is to use north south as one number line, or axis. Positive positions are north of the origin. Negative positions are south of the origin. The other axis goes east west. Positive positions on this axis are east of the origin. Negative positions are west of the origin. Pathfinder s exact position can be described with two numbers. These numbers are called coordinates. The graph at the left shows Pathfinder at the coordinates of (4, 2) m. The first number (or coordinate) gives the position on the east axis. Pathfinder is 4 m east of the origin. The second number gives the position on the north south axis. Pathfinder is 2 m north of the origin. axis - one of two (or more) number lines that form a graph. coordinates - values that give a position relative to an origin. Figure 4.8: A flat surface has two perpendicular dimensions: north south and east west. Each dimension has positive and negative directions. Maps A graph using north south and east west axes can accurately show where Pathfinder is. The graph can also show any path Pathfinder takes, curved or straight. This kind of graph is called a map. Many street maps use letters on the north south axis and numbers for the east west axis. For example, the coordinates F-4 identify the square that is in row F, column 4 of the map shown in Figure 4.9. Figure 4.9: Street maps often use letters and numbers for coordinates. 4.1 POSITION, SPEED, AND VELOCITY 83
Vectors on a map A trip with a turn Suppose you run east for 10 seconds at a speed of 2 m/s. Then you turn and run south at the same speed for 10 more seconds (Figure 4.10). Where are you compared to where you started? To get the answer, you figure out your east west changes and your north south changes separately. Figure each direction separately Your first movement has a velocity vector of +2 m/s on the east west axis. After 10 seconds your change in position is +20 meters (east). There are no more east west changes because your second movement is north south only. Your second movement has a velocity vector of 2 m/s north south. In 10 seconds you moved 20 meters. The negative sign means you moved south. Figuring your final position Now add up any east west changes to get your final east west position. Do the same for your north south position. Your new position is (+20 m, 20 m). 84 UNIT 2 MOTION AND FORCE Figure 4.10: A running trip with a turn. Captain Vector s Hidden Treasure Use these velocity vectors to determine the location of Captain Vector s hidden pirate treasure. Your starting place is (0, 0). 1. Walk at a velocity of 1 m/s south for 10 seconds. 2. Then, jog at a velocity of 3 m/s east for 5 seconds. 3. Run at a velocity of 5 m/s north for 2 seconds. 4. Then walk backward south at a velocity of 0.5 m/s for 2 seconds. Where is the treasure relative to your starting place?
Solving Problems: Velocity A train travels at 100 km/h heading east to reach a town in 4 hours. The train then reverses and heads west at 50 km/h for 4 hours. What is the train s position now? 1. Looking for: You are asked for position. 2. Given: You are given two velocity vectors and the times for each. 3. Relationships: change in position = velocity time 4. Solution: The first change in position is (+100 km/h) (4 h) = +400 km The second change in position is ( 50 km/h) (4 h) = 200 km The final position is (+400 km) + ( 200 km) = +200 km. The train is 200 km east of where it started. Your turn... a. You are walking around your town. First you walk north from your starting position and walk for 2 hours at 1 km/h. Then, you walk west for 1 hour at 1 km/h. Finally, you walk south for 1 hour at 2 km/h. What is your new position relative to your starting place? b. A ship needs to sail to an island that is 1,000 km south of where the ship starts. If the captain sails south at a steady velocity of 30 km/h for 30 hours, will the ship make it? MOTION Chapter 4 Fast Trains! The Bullet train of Japan was the world s first high-speed train. When it came into use in 1964, it went 210 km/h. Research today s high-speed trains of the world. How fast they go? Research to find out why the United States lags behind in having highspeed trains. Find out the advantages and disadvantages of having highspeed trains in the U.S. a. Your new position is 1kilometer west of where you started. b. No, because 30 km/h 30 h = 900 km. The island is still 100 km away. 4.1 POSITION, SPEED, AND VELOCITY 85.
Section 4.1 Review 1. What is the difference between distance and position? 2. From an origin you walk 3 meters east, 7 meters west, and then 6 meters east. Where are you now relative to the origin? 3. What is your average speed if you walk 2 kilometers in 20 minutes? 4. Give an example where instantaneous speed is different from average speed. 5. A weather report says winds blow at 5 km/h from the northeast. Is this description of the wind a speed or velocity? Explain your answer. 6. What velocity vector will move you 200 miles east in 4 hours traveling at a constant speed? 7. Give an example of a situation in which you would describe an object s position in: a. one dimension b. two dimensions c. three dimensions 8. A movie theater is 4 kilometers east and 2 kilometers south of your house. a. Give the coordinates of the movie theater. Your house is the origin. b. After leaving the movie theater, you drive 5 kilometers west and 3 kilometers north to a restaurant. What are the coordinates of the restaurant? Use your house as the origin. 86 UNIT 2 MOTION AND FORCE Look at the graphic below and answer the following questions. 1. How fast is each cyclist going in units of meters per second*? 2. Which cyclist is going faster? How much faster is this cyclist going compared to the other one? *The word per means for every or for each. Saying 5 kilometers per hour is the same as saying 5 kilometers for each hour. You can also think of per as meaning divided by. The quantity before the word per is divided by the quantity after it.