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Looking Back: Trigonometry In a previous course, you learned mathematical relationships among the sides and the angles of triangles. In this course you’ll use these relationships to analyze motion and other problems. © 2015 Pearson Education, Inc.

Chapter 1 Preview Stop to Think What is the length of the hypotenuse of this triangle? 6 cm 8 cm 10 cm 12 cm 14 cm Stop to Think Answer: C © 2015 Pearson Education, Inc.

Section 1.1 Motion: A First Look © 2015 Pearson Education, Inc.

Types of Motion Motion is the change of an object’s position or orientation with time. The path along which an object moves is called the object’s trajectory. © 2015 Pearson Education, Inc.

Making a Motion Diagram These motion diagrams in one dimension (straight- line) show objects moving at constant speed (skateboarder) speeding up (runner) slowing down (car). © 2015 Pearson Education, Inc.

Making a Motion Diagram This motion diagram shows motion in two dimensions with changes in both speed and direction. © 2015 Pearson Education, Inc.

QuickCheck 1.1 Car A Car B Motion diagrams are made of two cars. Both have the same time interval between photos. Which car, A or B, is going slower? Answer: A Car A Car B © 2015 Pearson Education, Inc.

QuickCheck 1.1 Car A Car B Motion diagrams are made of two cars. Both have the same time interval between photos. Which car, A or B, is going slower? Car A Car B © 2015 Pearson Education, Inc.

The Particle Model The particle model of motion is a simplification in which we treat a moving object as if all of its mass were concentrated at a single point © 2015 Pearson Education, Inc.

QuickCheck 1.2 Two runners jog along a track. The positions are shown at 1 s intervals. Which runner is moving faster? Runner A Runner B Answer: A © 2015 Pearson Education, Inc.

QuickCheck 1.2 Two runners jog along a track. The positions are shown at 1 s intervals. Which runner is moving faster? A Runner A Runner B © 2015 Pearson Education, Inc.

QuickCheck 1.3 Two runners jog along a track. The times at each position are shown. Which runner is moving faster? Runner A Runner B Both runners are moving at the same speed. Answer: C © 2015 Pearson Education, Inc.

QuickCheck 1.3 Two runners jog along a track. The times at each position are shown. Which runner is moving faster? Runner A Runner B Both runners are moving at the same speed. © 2015 Pearson Education, Inc.

Section 1.2 Position and Time: Putting Numbers on Nature © 2015 Pearson Education, Inc.

Position and Coordinate Systems To specify position we need a reference point (the origin), a distance from the origin, and a direction from the origin. The combination of an origin and an axis marked in both the positive and negative directions makes a coordinate system. © 2015 Pearson Education, Inc.

Position and Coordinate Systems The symbol that represents a position along an axis is called a coordinate. © 2015 Pearson Education, Inc.

Time For a complete motion diagram we need to label each frame with its corresponding time (symbol t) as read off a clock. © 2015 Pearson Education, Inc.

Changes in Position and Displacement A change of position is called a displacement. Displacement is the difference between a final position and an initial position: Δx = xf – xi = 0 – 50 ft = – 50 ft © 2015 Pearson Education, Inc.

Change in Time In order to quantify motion, we’ll need to consider changes in time, which we call time intervals. A time interval Δt measures the elapsed time as an object moves from an initial position xi at time ti to a final position xf at time tf. Δt is always positive. © 2015 Pearson Education, Inc.

Example 1.1 How long a ride? Carol is enjoying a bicycle ride on a country road that runs east- west past a water tower. Define a coordinate system so that increasing x means moving east. At noon, Carol is 3 miles (mi) east of the water tower. A half-hour later, she is 2 mi west of the water tower. What is her displacement over that half-hour? © 2015 Pearson Education, Inc.

Example 1.1 How long a ride? Carol is enjoying a bicycle ride on a country road that runs east- west past a water tower. Define a coordinate system so that increasing x means moving east. At noon, Carol is 3 miles (mi) east of the water tower. A half-hour later, she is 2 mi west of the water tower. What is her displacement during that half-hour? prepare Although it may seem like overkill for such a simple problem, you should start by making a drawing, with the x-axis along the road. © 2015 Pearson Education, Inc.

Example 1.1 How long a ride? (cont.) solve We’ve specified values for Carol’s initial and final positions in our drawing. We can thus compute her displacement: Δx = xf  xi = ( 2 mi)  (3 mi) =  5 mi assess Carol is moving to the west, so we expect her displacement to be negative—and it is. We can see from our drawing in Figure 1.15 that she has moved 5 miles from her starting position, so our answer seems reasonable. Carol travels 5 miles in a half-hour, quite a reasonable pace for a cyclist. © 2015 Pearson Education, Inc.

QuickCheck 1.4a Maria is at position x = 23 m. She then undergoes a displacement x = –50 m. What is her initial position? –27 m –50 m 23 m 73 m Answer: A © 2015 Pearson Education, Inc.

QuickCheck 1.4a Maria is at position x = 23 m. She then undergoes a displacement x = –50 m. What is her initial position? –27 m –50 m 23 m 73 m © 2015 Pearson Education, Inc.

QuickCheck 1.4b Maria is at position x = 23 m. She then undergoes a displacement x = –50 m. What is her final position? –27 m –50 m 23 m 73 m Answer: A © 2015 Pearson Education, Inc.

QuickCheck 1.4b Maria is at position x = 23 m. She then undergoes a displacement x = –50 m. What is her final position? –27 m –50 m 23 m 73 m © 2015 Pearson Education, Inc.

QuickCheck 1.5 An ant zig-zags back and forth on a picnic table as shown. The ant’s distance traveled and displacement are 50 cm and 50 cm 30 cm and 50 cm 50 cm and 30 cm 50 cm and –50 cm 50 cm and –30 cm Answer: E © 2015 Pearson Education, Inc. 31

QuickCheck 1.5 An ant zig-zags back and forth on a picnic table as shown. The ant’s distance traveled and displacement are 50 cm and 50 cm 30 cm and 50 cm 50 cm and 30 cm 50 cm and –50 cm 50 cm and –30 cm © 2015 Pearson Education, Inc. 32

Section 1.3 Velocity © 2015 Pearson Education, Inc.

Velocity and Speed Motion at a constant speed in a straight line is called uniform motion. © 2015 Pearson Education, Inc.

Example Problem Jane walks to the right at a constant rate, moving 3 m in 3 s. At t = 0 s she passes the x = 1 m mark. Draw her motion diagram from t = –1 s to t = 4 s. © 2015 Pearson Education, Inc.