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Chapter 2 Motion Along a Straight Line 2-0. Mathematical Concept 2.1. What is Physics? 2.2. Motion 2.3. Position and Displacement 2.4. Average Velocity and Average Speed 2.5. Instantaneous Velocity and Speed 2.6. Acceleration 2.7. Constant Acceleration: A Special Case 2.8. Another Look at Constant Acceleration 2.9. Free-Fall Acceleration 2.10. Graphical Integration in Motion Analysis Trigonometry Example 1 Using Trigonometric Functions On a sunny day, a tall building casts a shadow that is 67.2 m long. The angle between the sun’s rays and the ground is =50.0°, as Figure 1.6 shows. Determine the height of the building. Trigonometric Functions PYTHAGOREAN THEOREM Example 1 Using Trigonometric Functions On a sunny day, a tall building casts a shadow that is 67.2 m long. The angle between the sun’s rays and the ground is =50.0°, as Figure 1.6 shows. Determine the height of the building. What is the location of downtown Wilmington? Defining a Coordinate System One-dimensional coordinate system consists of: • a point of reference known as the origin (or zero point), • a line that passes through the chosen origin called a coordinate axis, one direction along the coordinate axis, chosen as positive and the other direction as negative, and the units we use to measure a quantity Scalars and Vectors • A scalar quantity is one that can be described with a single number (including any units) giving its magnitude. • A Vector must be described with both magnitude and direction. A vector can be represented by an arrow: •The length of the arrow represents the magnitude (always positive) of the vector. •The direction of the arrow represents the direction of the vector. A component of a vector along an axis (one-dimension) A UNIT VECTOR FOR A COORDINATE AXIS is a dimensionless vector that points in the direction along a coordinate axis that is chosen to be positive. A one-dimensional vector can be constructed by: •Multiply the unit vector by the magnitude of the vector •Multiply a sign: a positive sign if the vector points to the same direction of the unit vector; a negative sign if the vector points to the opposite direction of the unit vector. A component of a vector along an axis=sign × magnitude Difference between vectors and scalars • The fundamental distinction between scalars and vectors is the characteristic of direction. Vectors have it, and scalars do not. • Negative value of a scalar means how much it below zero; negative component of a vector means the direction of the vector points to a negative direction. Check Your Understanding 1 Which of the following statements, if any, involves a vector? (a) I walked 2 miles along the beach. (b) I walked 2 miles due north along the beach. (c) I jumped off a cliff and hit the water traveling at 17 miles per hour. (d) I jumped off a cliff and hit the water traveling straight down at 17 miles per hour. (e) My bank account shows a negative balance of –25 dollars. Motion • The world, and everything in it, moves. • Kinematics: describes motion. • Dynamics: deals with the causes of motion. One-dimensional position vector • The magnitude of the position vector is a scalar that denotes the distance between the object and the origin. • The direction of the position vector is positive when the object is located to the positive side of axis from the origin and negative when the object is located to the negative side of axis from the origin. Displacement • DISPLACEMENT is defined as the change of an object's position that occurs during a period of time. • The displacement is a vector that points from an object’s initial position to its final position and has a magnitude that equals the shortest distance between the two positions. • SI Unit of Displacement: meter (m) Example 2: Determine the displacement in the following cases: (a) A particle moves along a line from to (b) A particle moves from to (c) A particle starts at 5 m, moves to 2 m, and then returns to 5 m EXAMPLE 3: Displacements Three pairs of initial and final positions along an x axis represent the location of objects at two successive times: (pair 1) –3 m, +5 m; (pair 2) –3 m, –7 m; (pair 3) 7 m, –3 m. • (a) Which pairs give a negative displacement? • (b) Calculate the value of the displacement in each case using vector notation. Velocity and Speed A student standing still with the back of her belt at a horizontal distance of 2.00 m to the left of a spot of the sidewalk designated as the origin. A student starting to walk slowly. The horizontal position of the back of her belt starts at a horizontal distance of 2.47 m to the left of a spot designated as the origin. She is speeding up for a few seconds and then slowing down. Average Velocity • x2 and x1 are components of the position vectors at the final and initial times, and angle brackets denotes the average of a quantity. • SI Unit of Average Velocity: meter per second (m/s) Example 4 The World’s Fastest Jet-Engine Car Figure (a) shows that the car first travels from left to right and covers a distance of 1609 m (1 mile) in a time of 4.740 s. Figure (b) shows that in the reverse direction, the car covers the same distance in 4.695 s. From these data, determine the average velocity for each run. • Example 5: find the average velocity for the student motion represented by the graph shown in Fig. 2-9 between the times t1 = 1.0 s and t2 = 1.5 s. Average Speed Average speed is defined as: Check Your Understanding A straight track is 1600 m in length. A runner begins at the starting line, runs due east for the full length of the track, turns around, and runs halfway back. The time for this run is five minutes. What is the runner’s average velocity, and what is his average speed? EXAMPLE 6 You drive a beat-up pickup truck along a straight road for 8.4 km at 70 km/h, at which point the truck runs out of gasoline and stops. Over the next 30 min, you walk another 2.0 km farther along the road to a gasoline station. • (a) What is your overall displacement from the beginning of your drive to your arrival at the station? • (b) What is the time interval from the beginning of your drive to your arrival at the station? What is your average velocity from the beginning of your drive to your arrival at the station? Find it both numerically and graphically. Suppose that to pump the gasoline, pay for it, and walk back to the truck takes you another 45 min. What is your average speed from the beginning of your drive to your return to the truck with the gasoline? Instantaneous Velocity and Speed • The instantaneous velocity of an object can be obtained by taking the slope of a graph of the position component vs. time at the point associated with that moment in time • The instantaneous velocity can be obtained by taking a derivative with respect to time of the object's position. • Instantaneous speed, which is typically called simply speed, is just the magnitude of the instantaneous velocity vector, Example 7 The following equations give the position component, x(t), along the x axis of a particle's motion in four situations (in each equation, x is in meters, t is in seconds, and t > 0): (1) x = (3 m/s)t – (2 m); (2) x = (–4 m/s2)t2 – (2 m); (3) x = (–4 m/s2)t2; (4) x = –2 m. • (a) In which situations is the velocity of the particle constant? • (b) In which is the vector pointing in the negative x direction? How to Describe Change of Velocity ? Definition of Acceleration SI Unit of Average Acceleration: meter per second squared (m/s2) Instantaneous acceleration: • An object is accelerated even if all that changes is only the direction of its velocity and not its speed. • It is important to realize that speeding up is not always associated with an acceleration that is positive. Likewise, slowing down is not always associated with an acceleration that is negative. The relative directions of an object's velocity and acceleration determine whether the object will speed up or slow down. EXERCISE A cat moves along an x axis. What is the sign of its acceleration if it is moving (a) in the positive direction with increasing speed, (b) in the positive direction with decreasing speed, (c) in the negative direction with increasing speed, and (d) in the negative direction with decreasing speed? EXAMPLE 7: Position and Motion A particle's position on the x axis of Fig. 2-1 is given by with x in meters and t in seconds. • (a) Find the particle's velocity function and acceleration function . • (b) Is there ever a time when ? • (c) Describe the particle's motion for Constant Acceleration: A Special Case Free-Fall Acceleration Equations of Motion with Constant Acceleration Example 8 A Falling Stone A stone is dropped from rest from the top of a tall building, as Figure 2.17 indicates. After 3.00 s of free-fall, (a) what is the velocity of the stone? (b) what is the displacement y of the stone? Example 9 An Accelerating Spacecraft The spacecraft shown in Figure 2.14a is traveling with a velocity of +3250 m/s. Suddenly the retrorockets are fired, and the spacecraft begins to slow down with an acceleration whose magnitude is 10.0 m/s2. What is the velocity of the spacecraft when the displacement of the craft is +215 km, relative to the point where the retrorockets began firing Example 10 Spotting a police car, you brake your Porsche from a speed of 100 km/h to a speed of 80.0 km/h during a displacement of 88.0 m, at a constant acceleration. • What is that acceleration? • (b) How much time is required for the given decrease in speed? I Graphical Integration n in Motion Analysis t e g r a t i o n i n M o t i o n Conceptual Question 1. A honeybee leaves the hive and travels 2 km before returning. Is the displacement for the trip the same as the distance traveled? If not, why not? 2. Two buses depart from Chicago, one going to New York and one to San Francisco. Each bus travels at a speed of 30 m/s. Do they have equal velocities? Explain. 3. One of the following statements is incorrect. (a) The car traveled around the track at a constant velocity. (b) The car traveled around the track at a constant speed. Which statement is incorrect and why? 4. At a given instant of time, a car and a truck are traveling side by side in adjacent lanes of a highway. The car has a greater velocity than the truck. Does the car necessarily have a greater acceleration? Explain. 5. The average velocity for a trip has a positive value. Is it possible for the instantaneous velocity at any point during the trip to have a negative value? Justify your answer. 6. An object moving with a constant acceleration can certainly slow down. But can an object ever come to a permanent halt if its acceleration truly remains constant? Explain.

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instantaneous velocity, average speed, one dimension, instantaneous acceleration, initial velocity, time interval, acceleration due to gravity, Free Fall, Motion Along a Straight Line, negative acceleration

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posted: | 4/28/2011 |

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