Physics 101 Motion in One Dimension Prof. E. Aslฤฑ Yetkin Kinematics Describes motion while ignoring the external agents that might have caused or modified the motion. First we will be focusing motion in one dimension • Motion along a straight line Motion represents a continual change in object’s position. Types of Motion • Translational • Car traveling on a highway. • Rotational • Earth’s spin on it axis. • Vibrational • Back and forth movement of a pendulum. Particle Model • In this course, unless otherwise stated we will use particle model. • A particle is a point-like object; has mass but infinitesimal size. Position • The object’s position is its location with respect to a chosen reference point. • Reference point can be considered as the origin of the coordinate system. Position Consider a car moving back and forth along the x-axis. We are only interested in the car’s translational motion, so we will model car as a particle Pictorial Representation Position Tabular Representation Graphical Representation Displacement Given the data provided in Table 2.1, we can easily determine the change in position of the car for various time intervals. Displacement is defined as the change in position during some time interval. • Represented as Δ๐ฅ where Δ๐ฅ ≡ ๐ฅ๐ − ๐ฅ๐ • SI units are meters (m) • Δ๐ฅ can be positive or negative It is very important to recognize the difference between the displacement and distance traveled. Distance is the length of a path followed by a particle. Distance vs. Displacement Vectors and Scalars • Vector quantities need both magnitude (size or numerical value) and direction to completely described them. • We will use + and – signs to indicate vector directions in this chapter. • Scalar quantities are completely described by magnitude only. Average Velocity • The average velocity is rate at which the displacement occurs. • ๐ฃ๐ฅ,a๐ฃg ≡ Δ๐ฅ Δ๐ก = ๐ฅ๐ −๐ฅ๐ Δ๐ก • The x indicates motion is along the x-axis. • The dimensions are length/time [L/T] • The SI units are m/s • Since the time interval is always positive, the average velocity of a particle moving in 1D can be both positive or negative. Average Velocity The average velocity is also the slope of the line in the position-time graph. Average Speed • Speed is a scalar quantity. • โ๐๐ ๐กโ๐ ๐ ๐๐๐ ๐ข๐๐๐ก๐ ๐๐ ๐ฃ๐๐๐๐๐๐ก๐ฆ • Defined as total distance / total time: ๐ฃ๐๐ฃ๐ ≡ ๐ ๐ก • The speed has no direction and is always expressed as a positive number. • Neither average velocity nor average speed gives details about the trip described. Quick Quiz: Under which of the following conditions is the magnitude of average velocity of a particle moving in one-dimension smaller than the average speed over some time interval? a) A particle moves in the +x direction without reversing. b) A particle moves in the –x direction without reversing. c) A particle moves in the +x direction and then reverses the direction of its motion. d) There are no conditions for which is true. Average Speed & Average Velocity • The average speed is not the magnitude of the average velocity. • For example: A runner ends at her starting point • Her displacement is zero. • Therefore, her velocity is zero. • However, the distance she traveled is not zero, so the speed is not zero. Instantaneous Velocity Often, we need to know the velocity of a particle at a particular instant in time t rather than the average velocity over a finite time interval Δt The slope of the line 1: the average velocity for the interval during which car moved from A to B. The slope of the line 2: the average velocity for the interval during which car moved from A to F. Instantaneous Velocity Let’s focus on the interval during which car moved from A to B. While moving closer to point A, the line between the points become steeper and steeper. When the two points become extremely close together the line becomes a tangent to line to the curve. The slope of this tangent line represents the velocity of the car at point A. Instantaneous Velocity Instantaneous velocity ๐ฃ๐ฅ equals the limiting value of the ratio Δ ๐ฅΤΔ ๐ก as Δ๐ก approaches to zero. Δ๐ฅ ๐ฃ๐ฅ ≡ lim Δ๐ก→0 Δ๐ก In calculus notation, this limit is called the “derivative of ” x with respect to t. Δ๐ฅ โ ๐ฅ ๐ฃ๐ฅ ≡ lim = Δ๐ก→0 Δ๐ก โ ๐ก The instantaneous velocity can be positive, negative or zero. From here on, we use the word “velocity” to designate instantaneous velocity. Instantaneous Speed Instantaneous speed is the magnitude of the instantaneous velocity. The instantaneous speed has no direction associated with it. A Particle Under Constant Velocity Constant velocity indicates the instantaneous velocity at any instant during a time interval is the same as the average velocity during that time interval. ๐ฃ๐ฅ = ๐ฃ๐ฅ,๐๐ฃ๐ The mathematical representation of this situation is the equation. Δ๐ฅ ๐ฅ๐ − ๐ฅ๐ ๐ฃ๐ฅ = = Δ๐ก Δ๐ก or ๐ฅ๐ = ๐ฅ๐ + ๐ฃ๐ฅ Δ๐ก Common practice is to let ti = 0 and the equation becomes ๐ฅ๐ = ๐ฅ๐ + ๐ฃ๐ฅ ๐ก For constant velocity ๐ฃ๐ฅ A Particle Under Constant Velocity The graph represents the motion of a particle under constant velocity. The slope of the graph is the value of the constant velocity. The y-intercept is Xi. A Particle Under Constant Speed A particle under constant velocity moves with a constant speed along a straight line. A particle can also move with a constant speed along a curved path. The primary equation is the same as for average speed, with the average speed replaced by the constant speed. ๐ ๐ฃ= Δ๐ก Acceleration When the velocity of a particle changes with time, the particle is said to be accelerating. Suppose an object that can be modeled as particle moving along the x-axis has an initial velocity ๐ฃ๐ฅ๐ at time ti at position A and a final velocity ๐ฃ๐ฅ๐ at time tf at position B as in figure. The average acceleration ๐๐ฅ,๐๐ฃ๐ of the particle is defined as the change in velocity Δ๐๐ฅ divided by the time interval Δt during which that change occurs. ๐๐ฅ,๐๐ฃ๐ Δ๐๐ฅ ๐ฃ๐ฅ๐ − ๐ฃ๐ฅ๐ ≡ = Δ๐ก ๐ก๐ − ๐ก๐ Average Acceleration ๐๐ฅ,๐๐ฃ๐ Δ๐๐ฅ ๐ฃ๐ฅ๐ − ๐ฃ๐ฅ๐ ≡ = Δ๐ก ๐ก๐ − ๐ก๐ Dimensions are L/T2 SI units are m/s2 In one dimension, positive and negative can be used to indicate direction. Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as Δ๐ก approaches 0. Δ๐ฃ๐ฅ ๐๐ฃ๐ฅ ๐ 2 ๐ฅ ๐๐ฅ = lim = = 2 Δ๐ก→0 Δ๐ก ๐๐ก ๐๐ก The term acceleration will mean instantaneous acceleration. If average acceleration is wanted, the word average will be included. Instantaneous Acceleration The instantaneous acceleration is the limit of the average acceleration as Δ๐ก approaches 0. 2 Δ๐ฃ๐ฅ ๐๐ฃ๐ฅ ๐ ๐ฅ ๐๐ฅ = lim = = 2 Δ๐ก→0 Δ๐ก ๐๐ก ๐๐ก The term acceleration will mean instantaneous acceleration. If average acceleration is wanted, the word average will be included. Graphical Relationships Between x, vx, and ax When a position-time graph is given for an object, the corresponding velocity-time and acceleration-time plots can be plotted. The velocity-time graph is found by measuring the slope of the position-time graph at every instant. The acceleration-time graph is found by measuring the slope of the velocity-time graph at every instant. Acceleration and Velocity, Directions When an object’s velocity and acceleration are in the same directions, that object is speeding up. When an object’s velocity and acceleration are in the opposite direction, the object is slowing down. Negative acceleration does not necessarily mean the object is slowing down. • If the acceleration and velocity are both negative, the object is speeding up. The word deceleration has the connotation of slowing down. • This word will not be used in the text. Motion Diagrams A motion diagram can be formed by imagining the stroboscope photograph of a moving object. • Red arrows represent the velocity • Purple arrows represent acceleration A Particle Under Constant Acceleration If acceleration is constant, the average acceleration ax,avg over any time interval is numerically equal to instantaneous acceleration ax at any instant within the time interval, and the velocity changes at the same rate through out the motion. ๐๐ฅ,๐๐ฃ๐ Δ๐ฃ๐ฅ ๐ฃ๐ฅ๐ − ๐ฃ๐ฅ๐ = ๐๐ฅ = = Δ๐ก ๐ก๐ − ๐ก๐ Assuming ti = 0, ๐ฃ๐ฅ๐ = ๐ฃ๐ฅ๐ + ๐๐ฅ ๐ก (for constant acceleration) A Particle Under Constant Acceleration For constant acceleration, ๐ฃ๐ฅ,๐๐ฃ๐ ๐๐ฅ๐ + ๐๐ฅ๐ = 2 The average velocity can be expressed as the arithmetic mean of the initial and final velocities • This applies only in the situations where acceleration is constant. A Particle Under Constant Acceleration For constant acceleration, 1 ๐ฅ๐ = ๐ฅ๐ + ๐ฃ๐ฅ,๐๐ฃ๐ ๐ก = ๐ฅ๐ + ๐ฃ๐ฅ๐ + ๐ฃ๐ฅ๐ ๐ก 2 This gives you the position of the particle in terms of time and velocities. Does not give you the acceleration. 1 ๐ฅ๐ = ๐ฅ๐ + ๐๐ฅ๐ ๐ก + ๐๐ฅ๐ก 2 2 Gives the final position in terms of initial velocity and acceleration. Does not tell you information about final velocity. A Particle Under Constant Acceleration For constant acceleration, 2 ๐ฃ๐ฅ๐ = 2 ๐ฃ๐ฅ๐ + 2๐๐ฅ (๐ฅ๐ − ๐ฅ๐ ) This gives final velocity in terms of acceleration and displacement. Does not tell you any information about the time. When acceleration is zero a=0 • ๐ฃ๐ฅ๐ = ๐ฃ๐ฅ๐ = ๐ฃ๐ฅ • ๐ฅ๐ = ๐ฅ๐ + ๐ฃ๐ฅ ๐ก Constant acceleration model reduces to the constant velocity model. Kinematic Equations Summary Graphical Look at Motion: Displacement – Time curve • The slope of the curve is the velocity. • The curved line indicates the velocity is changing. Therefore, there is an acceleration. Graphical Look at Motion: Velocity – Time curve • The slope gives the acceleration. • The straight line indicates a constant acceleration. Graphical Look at Motion: Acceleration – Time curve • The zero slope indicates a constant acceleration.
