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Motion Along a Line: Position, Displacement, Speed, Velocity, and Acceleration

This chapter introduces the concepts of position, displacement, speed, velocity, and acceleration in one-dimensional motion. It covers kinematics and dynamics in order to develop a comprehensive understanding of motion.

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Motion Along a Line: Position, Displacement, Speed, Velocity, and Acceleration

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  1. Chapter 2 Motion Along a Line

  2. Motion Along a Line • Position & Displacement • Speed & Velocity • Acceleration • Describing motion in 1D • Free Fall Ch_02b-Revised 5/31/2010

  3. Introduction • Kinematics - Concepts needed to describe motion - displacement, velocity & acceleration. • Dynamics - Deals with the effect of forces on motion. • Mechanics - Kinematics + Dynamics Ch_02b-Revised 5/31/2010

  4. Goals of Chapter 2 Develop an understanding of kinematics that comprehends the interrelationships among • physical intuition • equations • graphical representations When we finish this chapter you should be able to move easily among these different aspects. Ch_02b-Revised 5/31/2010

  5. Kinematic Quantities Overview The words speed and velocity are used interchangably in everyday conversation but they have distinct meanings in the physics world. Ch_02b-Revised 5/31/2010

  6. 0 x1 0 x2 Position & Displacement The position (x) of an object describes its location relative to some origin or other reference point. The position of the red ball differs in the two shown coordinate systems. Ch_02b-Revised 5/31/2010

  7. x (cm) 1 2 0 1 2 The position of the ball is The + indicates the direction to the right of the origin. Ch_02b-Revised 5/31/2010

  8. x (cm) 1 1 2 0 2 The position of the ball is The  indicates the direction to the left of the origin. Ch_02b-Revised 5/31/2010

  9. The displacement is the change in an object’s position. It depends only on the beginning and ending positions. All Δ quantities will have the final value 1st and the inital value last. Ch_02b-Revised 5/31/2010

  10. x (cm) 1 2 0 1 2 Example: A ball is initially at x = +2 cm and is moved to x = -2 cm. What is the displacement of the ball? Ch_02b-Revised 5/31/2010

  11. Example: At 3 PM a car is located 20 km south of its starting point. One hour later its is 96 km farther south. After two more hours it is 12 km south of the original starting point. (a) What is the displacement of the car between 3 PM and 6 PM? xi = –20 km and xf = –12 km Use a coordinate system where north is positive. Ch_02b-Revised 5/31/2010

  12. Example continued (b) What is the displacement of the car from the starting point to the location at 4 pm? xi = 0 km and xf = –96 km (c) What is the displacement of the car from 4 PM to 6 PM? xi = –96 km and xf = –12 km Ch_02b-Revised 5/31/2010

  13. Velocity: Rate of Change of Position Velocity is a vector that measures how fast and in what direction something moves. Speed is the magnitude of the velocity. It is a scalar. Ch_02b-Revised 5/31/2010

  14. In 1-dimension the average velocity is vav is the constant speed and direction that results in the same displacement in a given time interval. Ch_02b-Revised 5/31/2010

  15. x(m) x2 x1 t2 t1 t (sec) On a graph of position versus time, the average velocity is represented by the slope of a chord. Ch_02b-Revised 5/31/2010

  16. x (m) t (sec) This is represented by the slope of a line tangent to the curve on the graph of an object’s position versus time. Ch_02b-Revised 5/31/2010

  17. vx (m/s) t (sec) The area under a velocity versus time graph (between the curve and the time axis) gives the displacement in a given interval of time. Ch_02b-Revised 5/31/2010

  18. Example (text problem 2.11): Speedometer readings are obtained and graphed as a car comes to a stop along a straight-line path. How far does the car move between t = 0 and t = 16 seconds? Since there is not a reversal of direction, the area between the curve and the time axis will represent the distance traveled. Ch_02b-Revised 5/31/2010

  19. Example continued: The rectangular portion has an area of Lw = (20 m/s)(4 s) = 80 m. The triangular portion has an area of ½bh = ½(8 s) (20 m/s) = 80 m. Thus, the total area is 160 m. This is the distance traveled by the car. Ch_02b-Revised 5/31/2010

  20. The Most Important Graph- V vs T The values of the curve gives the instantaneous VELOCITY. The slope of the curve gives the ACCELERATION. Negative areas are possible. Area under the curve gives DISTANCE. Ch_02b-Revised 5/31/2010

  21. Acceleration: Rate of Change of Velocity These have interpretations similar to vav and v. Ch_02b-Revised 5/31/2010

  22. Example (text problem 2.29): The graph shows speedometer readings as a car comes to a stop. What is the magnitude of the acceleration at t = 7.0 s? The slope of the graph at t = 7.0 sec is Ch_02b-Revised 5/31/2010

  23. Motion Along a Line With Constant Acceleration For constant acceleration the kinematic equations are: Also: Ch_02b-Revised 5/31/2010

  24. A Modified Set of Equations For constant acceleration the kinematic equations are: Also: Ch_02b-Revised 5/31/2010

  25. Visualizing Motion Along a Line with Constant Acceleration Motion diagrams for three carts: Ch_02b-Revised 5/31/2010

  26. Graphs of x, vx, ax for each of the three carts Ch_02b-Revised 5/31/2010

  27. Free Fall A stone is dropped from the edge of a cliff; if air resistance can be ignored, we say the stone is in free fall. The magnitude of the acceleration of the stone is afree fall = g = 9.80 m/s2, this acceleration is always directed toward the Earth. The velocity of the stone changes by 9.8 m/severy sec. Ch_02b-Revised 5/31/2010

  28. Free Fall Assumption: acceleration due to gravity is g g = 9.8 m/s2 ≈ 10 m/s2 Ch_02b-Revised 5/31/2010

  29. y viy x ay Example: You throw a ball into the air with speed 15.0 m/s; how high does the ball rise? Given: viy = +15.0 m/s; ay = 9.8 m/s2 To calculate the final height, we need to know the time of flight. Time of flight from: Ch_02b-Revised 5/31/2010

  30. Example continued: The ball rises untilvfy = 0. The height: Ch_02b-Revised 5/31/2010

  31. y x 369 m Example (text problem 2.45): A penny is dropped from the observation deck of the Empire State Building 369 m above the ground. With what velocity does it strike the ground? Ignore air resistance. Given: viy = 0 m/s, ay = 9.8 m/s2, y = 369 m Unknown: vfy ay Use: Ch_02b-Revised 5/31/2010

  32. Example continued: (downward) How long does it take for the penny to strike the ground? Given: viy= 0 m/s, ay = 9.8 m/s2, y = 369 m Unknown: t Ch_02b-Revised 5/31/2010

  33. Summary • Position • Displacement Versus Distance • Velocity Versus Speed • Acceleration • Instantaneous Values Versus Average Values • The Kinematic Equations • Graphical Representations of Motion • Free Fall Ch_02b-Revised 5/31/2010

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