Chapter 1 Problem 28: Agenda. Quantities in Motion. Displacement Isn t Distance. Velocity. Speed 1/23/14

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1 Agenda We need a note-taker! If you re interested, see me after class. Today: HW Quiz #1, 1D Motion Lecture for this week: Chapter 2 (finish reading Chapter 2 by Thursday) Homework #2: continue to check the class website/schedule page Chapter 1 Problem 28: A city has streets laid out in a square grid, with each block 135 m long. If you drive north for three blocks, then west for three blocks, how far are you from your starting point? Show all your work. Quantities in Motion Any motion involves three concepts Displacement (x, m) Velocity (v, m/s) Acceleration (a, m/s 2 ) These concepts can be used to study objects in motion Displacement Isn t Distance The displacement of an object is not the same as the distance it travels Example: Throw a ball straight up and then catch it at the same point you released it The distance is twice the height The displacement is zero Speed The average speed of an object is defined as the total distance traveled divided by the total time elapsed total distance Average speed = total time v = d t Speed is a scalar quantity Velocity It takes time for an object to undergo a displacement The average velocity is rate at which the displacement occurs v average = Δx Δt = x x f i t f t i generally use a time interval, so let t i = 0 1

2 Velocity continued Direction will be the same as the direction of the displacement (time interval is always positive) + or - is sufficient Units of velocity are m/s Speed vs. Velocity Cars on both paths have the same average velocity since they had the same displacement in the same time interval The car on the blue path will have a greater average speed since the distance it traveled is larger Distance & Displacement Lecture- Tutorial Work with a partner or two Read directions and answer all questions carefully. Take time to understand it now! Come to a consensus answer you all agree on before moving on to the next question. If you get stuck, ask another group for help. If you get really stuck, raise your hand and I will come around. Graphical Interpretation of Velocity Velocity can be determined from a position-time graph Average velocity equals the slope of the line joining the initial and final positions An object moving with a constant velocity will have a graph that is a straight line Average Velocity, Constant Average Velocity, Non Constant The straight (not curved) line indicates constant velocity The slope of the line is the value of the average velocity This motion is non-constant velocity The average velocity is the slope of the blue line joining two points 2

3 Instantaneous Velocity The limit of the average velocity as the time interval becomes infinitesimally short, or as the time interval approaches zero lim Δx v Δt 0 Δt The instantaneous velocity indicates what is happening at every point of time Instantaneous Velocity on a Graph The slope of the line tangent to the position-vs.-time graph is defined to be the instantaneous velocity at that time The instantaneous speed is defined as the magnitude of the instantaneous velocity Uniform Velocity Uniform velocity constant velocity The instantaneous velocities are always the same All the instantaneous velocities will also equal the average velocity Acceleration Changing velocity (non-uniform) means an acceleration is present Acceleration is the rate of change of the velocity a = Δv Δt = v v f i t f t i Units are m/s² Agenda Today: Finish Chapter 2 (Motion graphs, kinematics equations, and free fall) Reading for next week: Chapter 3 Still no change in enrollment HW #1/quiz can be picked up at the end of class Acceleration Vector quantity: rate of change of velocity When the sign of the velocity and the acceleration are the same (either positive or negative), then the speed is increasing When the sign of the velocity and the acceleration are in the opposite directions, the speed is decreasing 3

4 Relationship Between Acceleration and Velocity Relationship Between Velocity and Acceleration Uniform velocity (shown by red arrows maintaining the same size) Acceleration equals zero Velocity and acceleration are in the same direction Acceleration is uniform (blue arrows maintain the same length) Velocity is increasing (red arrows are getting longer) Positive velocity and positive acceleration Relationship Between Velocity and Acceleration Acceleration and velocity are in opposite directions Acceleration is uniform (blue arrows maintain the same length) Velocity is decreasing (red arrows are getting shorter) Velocity is positive and acceleration is negative Acceleration Lecture-Tutorial Work with a partner or two Read directions and answer all questions carefully. Take time to understand it now! Come to a consensus answer you all agree on before moving on to the next question. If you get stuck, ask another group for help. If you get really stuck, raise your hand and I will come around. The three graphs in the figure represent the position vs. time for objects moving along the x-axis. Which, if any, of these graphs is not physically possible? Graphical Interpretation of Acceleration Average acceleration is the slope of the line connecting the initial and final velocities on a velocity-time graph Instantaneous acceleration is the slope of the tangent to the curve of the velocitytime graph 4

5 Average Acceleration Parts (a), (b), and (c) of the figure represent three graphs of the velocities of different objects moving in straight-line paths as functions of time. The possible accelerations of each object as functions of time are shown in parts (d), (e), and (f). Match each velocity vs. time graph with the acceleration vs. time graph that best describes the motion. Kinematic Equations Used in situations with uniform acceleration v = v o + at Δx = vt = 1 ( 2 v + v o ) t Δx = v o t at Δx = vot + at 2 Gives displacement as a function of time, velocity and acceleration asked to find the final velocity v 2 = v o 2 + 2aΔx v = v o + at Shows velocity as a function of acceleration and time asked to find the displacement & vo + v f # Δx = vaverage t = $! t % 2 " Gives displacement as a function of velocity and time asked for the acceleration Careful! v avg is not the same as v i or v f 5

6 v 2 = v o 2 + 2aΔx Gives velocity as a function of acceleration and displacement asked for the time Free Fall All objects moving under the influence of gravity only are said to be in free fall Free fall does not depend on the object s original motion All objects falling near the earth s surface fall with a constant acceleration The acceleration is called the acceleration due to gravity, and indicated by g Acceleration due to Gravity Symbolized by g g = 9.80 m/s² When estimating, use g 10 m/s 2 g is always directed downward toward the center of the earth Ignoring air resistance and assuming g doesn t vary with altitude over short vertical distances, free fall is constantly accelerated motion Free Fall an object dropped Initial velocity is zero Let up be positive Use the kinematic equations Generally use y instead of x since vertical Acceleration is -g = m/s 2 v o = 0 a = -g 6