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When you decide where to place the measuring tape and when to start the stopwatch, you are defining a coordinate system, which tells you the location of the zero point of the variable you are studying and the direction in which the values of the variable increase The origin is the point at which both variables have the value zero In the example of the runner, the origin, represented by the zero end of the measuring tape, could be placed 6 m to the left of the tree The motion is in a straight line; thus, your measuring tape should lie along that straight line The straight line is an axis of the coordinate system You probably would place the tape so that the meter scale increases to the right of the zero, but putting it in the opposite direction is equally correct In Figure 2-6a, the origin of the coordinate system is on the left You can indicate how far away the runner is from the origin at a particular time on the simplified motion diagram by drawing an arrow from the origin to the point representing the runner, as shown in Figure 2-6b This arrow represents the runner s position, the separation between an object and the origin The length of the arrow indicates how far the object is from the origin, or the object s distance from the origin The arrow points from the origin to the location of the moving object at a particular time
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coordinate system origin position distance magnitude vectors scalars resultant time interval displacement
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Figure 2-6 In these motion diagrams, the origin is at the left (a), and the positive values of distance extend horizontally to the right The two arrows, drawn from the origin to points representing the runner, locate his position at two different times (b)
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Is there such a thing as a negative position Suppose you chose the coordinate system just described, placing the origin 4 m left of the tree with the d-axis extending in a positive direction to the right A position 9 m to the left of the tree, 5 m left of the origin, would be a negative position, as shown in Figure 2-7 In the same way, you could discuss a time before the stopwatch was started
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Figure 2-7 The arrow drawn on this motion diagram indicates a negative position
Vectors and scalars Quantities that have both size, also called magnitude, and direction, are called vectors, and can be represented by arrows Quantities that are just numbers without any direction, such as distance, time, or temperature, are called scalars This textbook will use boldface letters to represent vector quantities and regular letters to represent scalars You already know how to add scalars; for example, 06 02 08 How do you add vectors Think about how you would solve the following problem Your aunt asks you to get her some cold medicine at the store nearby You walk 05 km east from your house to the store, buy the cold medicine, and then walk another 02 km east to your aunt s house How far from the origin are you at the end of your trip The answer, of course, is 05 km east 02 km east 07 km east You also could solve this problem graphically, using the following method Using a ruler, measure and draw each vector The length of a vector should be proportional to the magnitude of the quantity being represented, so you must decide on a scale for your drawing For example, you might let 1 cm on paper represent 01 km The important thing is to choose a scale that produces a diagram of reasonable size with a vector that is about 5 10 cm long The vectors representing the two segments that made up your trip to your aunt s house are shown in Figure 2-8, drawn to a scale of 1 cm, which represents 01 km The vector that represents the total of these two, shown here with a dotted line, is 7 cm long According to the established scale, you were 07 km from the origin at the end of your trip The vector that represents the sum of the other two vectors is called the resultant The resultant always points from the tail of the first vector to the tip of the last vector
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