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The symbol means that the left-hand side of the equation is defined by the right-hand side It is a common misconception to say that the slope of a position-time graph gives the speed of the object Consider the slope of the position-time graph shown in Figure 2-21 The slope of this position-time graph is 50 m/s As you can see the slope indicates both the magnitude and direction Recall that average velocity is a quantity that has both magnitude and direction The slope of a position-time graph indicates the average velocity of the object and not its speed Take another look at Figure 2-21 The slope of the graph is 50 m/s and thus, the object has a velocity of 50 m/s The object starts out at a positive position and moves toward the origin It moves in the negative direction at a rate of 50 m/s Average speed The absolute value of the slope of a position-time graph tells you the average speed of the object; that is, how fast the object is moving The sign of the slope tells you in what direction the object is moving The combination of an object s average speed, v, and the direction in which it is moving is the average velocity, v Thus, for the object represented in Figure 2-21, the average velocity is 50 m/s, or 50 m/s in the negative direction Its average speed is 50 m/s Remember that if an object moves in the negative direction, then its displacement is negative This means that the object s velocity always will have the same sign as the object s displacement As you consider other types of motion to analyze in future chapters, sometimes the velocity will be the 4 5 important quantity to consider, while at other times, the speed will be the important quantity Therefore, it is a good idea to make sure that you understand the differences between velocity and speed so that you will be sure to use the right one later
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Figure 2-21 The object whose motion is represented here is moving in the negative direction at a rate of 50 m/s
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20 15 10 Position (m) 5 0 1 5 10 15 2 3 Time (s)
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2 Representing Motion
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The graph at the right describes the motion of a student riding his skateboard along a smooth, pedestrian-free sidewalk What is his average velocity What is his average speed
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Identify the coordinate system of the graph
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Position (m)
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Unknown:
v v
Find the average velocity using two points on the line v
d t d2 t2 d1 t1 60 m 40 s
Substitute d2 120 m, d1 Use magnitudes with signs indicating directions
10 20 30 40 50 60 70 80 90 Time (s)
120 m 80 s
60 m, t2
80 s, t1
40 s
Math Handbook
Slope page 850
15 m/s in the positive direction
The average speed, v , is the absolute value of the average velocity, or 15 m/s
Evaluate the Answer
Are the units correct m/s are the units for both velocity and speed Do the signs make sense The positive sign for the velocity agrees with the coordinate system No direction is associated with speed
a What is the ship s average speed b What is its average velocity 26 Describe, in words, the motion of the cruise ship in the previous problem 27 The graph in Figure 2-23 represents the motion of a bicycle Determine the bicycle s average speed and average velocity, and describe its motion in words 28 When Marilyn takes her pet dog for a walk, the dog walks at a very consistent pace of 055 m/s Draw a motion diagram and position-time graph to represent Marilyn s dog walking the 198-m distance from in front of her house to the nearest fire hydrant
Position (m)
25 The graph in Figure 2-22 describes the motion of a cruise ship during its voyage through calm waters The positive d-direction is defined to be south
Time (s) 1 0 2 3 4
2 Figure 2-22 20 Position (km) 15 10 5 0 Figure 2-23 5 10 15 20 25 30 Time (min)
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