Lecture 3: 1-D Kinematics This Week’s Announcements: • Class Webpage: • http://kestrel.nmt.edu/~dmeier/phys121/phys121.html • visit regularly • Our TA is Lorraine Bowman Week 2 Reading: Chapter 2 - Giancoli Due: 9/1 Week 2 Assignments: - HW #1 (chp 1: Q5, P2, P8, P39 & chp 2: Q2, Q18, P7, P9a-d, P35, P42) - for this one assignment you must report your answers with proper units and sig. figs. for full credit !! - MasteringPhysics: -Introduction to Mastering Physics (if not done yet) This Week’s Goal: - Define and introduce the basic kinematic variables, such as displacement, velocity and acceleration, and begin applying them to one dimensional motion. Wee! I’m learning kinematics space (x) Space-time diagrams time - Graph shows the evolution of the position (1-D) in time (how things move in time) --- not (x vs. y) but x vs. time Distance and Displacement + - Position: The spatial coordinate of an object as a function of time [x]. -generally have the freedom to select the zero of our coordinate system as we like - Distance: The length covered along a path. Xto = +1 mil -always a positive quantity xfrom = -1 mil - Displacement: The net change in position (and direction) for a path [Δx = x - xo]. -Displacement of a round trip is zero ! X =0 - - Keep to 1-D for the moment: 1 mil space (x) Introduction to space-time diagrams 20 m 30 m - Space-time diagram for trip to Little Caesar’s -(walking at a constant rate, and waiting in line) 50 m time Clicker Question: 3) Which of the following space-time diagrams represent an object which ultimately comes to a stop and remains stopped ? x a) x t b) x t c) x t d) x t e) t Clicker Question: 3) Which of the following space-time diagrams represent an object which ultimately comes to a stop and remains stopped ? x a) x t b) x t c) x t d) x t e) t Velocity and Speed - Average speed, s: (distance traveled)/(total time) x Slope = rise/run = Δx/Δt x2 Δx = x2 - x1 -always a positive quantity x1 - Average velocity: (displacement)/(total time) -includes direction information t1 t2 Δt = t2 - t1 t x - instantaneous velocity, v, (or simply “velocity”): (The change in displacement over a very tiny time interval) /(that tiny time interval) dx dt -includes direction information x(t+h) - x(t) = Δx - instantaneous speed, v, (or simply “speed”): The magnitude of the instantaneous velocity. -always a positive quantity h = Δt t Slope = rise/run = Δx/Δt V = lim h 0 x(t+h) - x(t) h V = dx/dt -This is precisely the definition of the time derivative of x. Calculus Digression x(t) = t dx(t)/dt =1 Slope = 2/2 = 1 * Slope = 2/2 = 1 * 2 2 * Slope = rise/run Slope over infinitesimal distance=dx(t)/dt 2 V = lim h 0 x(t+h) - x(t) h 2 x(t) = t2 2 2 = dx dt dx(t)/dt =2t Slope = 2/2 = 1 Slope = 4/1 = 4 - When the curve is horizontal, Δx is zero, so the slope Δx/Δt = 0 -setting the derivative to zero finds the local minima or maxima ∫ = integral = “anti-derivative” * 4 1 -2 * Slope = -2/1 = -2 1 2 * * 1 Slope = 0/1 = 0 1 Slope = 2/1 = 2 Clicker Question: 4) The following is a space time diagram for an object moving in 1-D: x t Which of the following represents the velocity vs. time diagram for the object? v v t a) v t b) v t c) v t d) t e) Clicker Question: 4) The following is a space time diagram for an object moving in 1-D: x t Which of the following represents the velocity vs. time diagram for the object? v v t a) v t b) v t c) v t d) t e) Acceleration - Acceleration is the change in velocity with time, in a completely analogous fashion to velocity being a change in displacement with time v dv dt - average acceleration: (change in velocity)/(total time) v(t+h) - v(t) -includes direction information - instantaneous acceleration, a, (or simply “acceleration”): (The change in velocity over a very tiny time interval) /(that tiny time interval) -includes direction information h v(t+h) - v(t) a = lim h 0 h t = d 2x dv dt = dt2 = a -This is precisely the definition of the time derivative of v. Velocity x (slope on space-time diagram) dx dt Acceleration (slope on velocity-time diagram) v t dv dt t Clicker Question: 5) The following is a velocity vs. time diagram for an object moving in 1-D: v t Which of the following represents the acceleration vs. time diagram for the object? a a t a) a t b) a t c) a t d) t e) Clicker Question: 5) The following is a velocity vs. time diagram for an object moving in 1-D: v t Which of the following represents the acceleration vs. time diagram for the object? a a t a) a t b) a t c) a t d) t e) x t Example: Non constant acceleration v t a t Clicker Question: x 6) Which of the following represents the time rate of change of the acceleration (d3x/dt3) [aka the ‘jerk’]? v + (d3x/dt3) t - e) (d3x/dt3) + - t t + a (d3x/dt3) d) t t - - c) (d3x/dt3) + b) (d3x/dt3) - a) + t t t Clicker Question: x 6) Which of the following represents the time rate of change of the acceleration (d3x/dt3) [aka the ‘jerk’]? v + (d3x/dt3) t - e) (d3x/dt3) + - t t + a (d3x/dt3) d) t t - - c) (d3x/dt3) + b) (d3x/dt3) - a) + t t t Motion diagrams 1-D: velocity - speeding up: acceleration present x - acceleration sign (+) if velocity increases in direction of increasing coordinate - slowing down: acceleration present velocity x - A graph the represents the location of an object at a series of equal spaced steps in time - acceleration sign (-) if velocity decreases in direction of increasing coordinate 2-D: y velocity x General Definitions: x(t) = the position of an object as a function of time v(t) = a(t) = dv dt = dx dt = ( ) dt dt d dx -Orv = ∫ a dt x = ∫ v dt = ∫ [∫a dt] dt d 2x dt2