1.7 POSITION, VELOCITY & Learning Objectives: How are motion graphs ACCELERATION VS TIME interpreted? GRAPHS (Guided Notes) 1.3 Create and interpret graphs of 1-dimensional motion, such as position vs. time, distance vs. time, speed vs. time, velocity vs. time, and acceleration vs. time where acceleration is constant. Understanding the definitions of velocity and acceleration allows us to create graphs which illustrate the motion. The mathematical definitions are as follows: x x x v v v final initial final initial t t final tinitial t t final tinitial These definitions look a lot like the equation for the slope of a line used in math. This is because it is the slope of a line. The slope equation is: Slope of a line = VELOCITY: First recall how VELOCITY IS CALCULATED: tinitial = 5 sec tfinal = 25 sec Xinitial = 50 meters Xfinal = 450 meters Plot the points on the graph and sketch the line created by the points. Estimate the points (This does not have to be accurate) Label each axis of the graph and give the graph a title. _____ vs ______ Time Postion (seconds) (meters) ti = xi = tf = xf = How is velocity represented on the graph? From this graph we can see that the position is: ACCELERATION: Remember how ACCELERATION IS CALCULATED? tinitial = 5 sec tfinal = 15 sec Vinitial = 50 meters/sec Xfinal = 250 meters/sec Plot the points on the graph and sketch the line created by the points. Label the axis of the graph and give it a title. ________ vs _________ Time Velocity (seconds) (m/s) ti = vi = tf = vf = How is the acceleration represented on the graph? What does it mean if acceleration is positive? The velocity of the object is (speeding up or slowing down) The acceleration vector and the velocity are pointing in ( the same or opposite) directions. tinitial = 9 sec tfinal = 12 sec ACCELERATION OR DECELERATION? Vinitial = 28 meters/sec Xfinal = 13 meters/sec Plot the points on the graph and sketch the line created by the points. Label the axis of the graph and give it a title. Time Velocity (seconds) (m/s) ti = vi = tf = vf = What does it mean if acceleration is negative? The velocity of the object is ___________________________ The acceleration vector and the velocity are ______________________ LAST NAME _____________________ FIRST NAME _____________________ DATE___________ PERIOD____ 1.7 GRAPHICAL ANALYSIS OF MOTION (STUDENT EXERCISE) ANALYZING THE POSITION VS TIME GRAPH (x vs t) Look at the graph above which represents the position of the biker with respect to elapsed time. The numbers (1, 2, 3) represent 3 sections of the graph with different slopes. How far from his starting point is the biker at 800 seconds? at 600 seconds? Does the biker make it back to his original position after the entire time represented on the graph? Calculate the slope of section 1: What is the velocity during the first 600 seconds? What is the slope of section 2? What is the velocity from 600 to 1000 seconds? Calculate the slope of section 3: What is the velocity from 1000 to 1800 seconds? ANOTHER X VS T GRAPH In this graph the position of the object is _____________ at a(n) ________________ rate. This means that it is moving faster and faster or ______________ which means its ____________ is increasing. The slope of this graph represents the _________. FREE FALLING BODY What happens to the velocity of an object that is allowed to fall, (This is called a Free Falling Body)? As the object falls the velocity will____________ at a _______________ rate. What happens to the position of an object that is allowed to fall, (This is called a Free Falling Body)? As the object falls the position will____________ at a _______________ rate. What will the following graphs look like: X vs t V vs t v v v final initial t t t x (m) final initial v (m/s) t (sec) t (sec) ACCELERATION FROM THE VELOCITY GRAPH The slope of the v vs t graph is the acceleration. Here the velocity of the object is ____________ at a(n) __________ rate. Calculate the slope of the graph to find the ____________ LET’S SEE HOW THE ONE GRAPH CREATES THE NEXT. EXAMPLE: X IS CONSTANT, OBJECT STAYS IN SAME PLACE X vs t The x vs t graph is constant so the v vs t graph will be (positive/negative/zero) x (m) t (sec) V vs t The v vs t graph is constant so the a vs t graph will be (positive/zero/negative) v (m/s) t (sec) a vs t a (m/s) t (sec) EXAMPLE: X CHANGING, OBJECT IS MOVING X vs t The x vs t graph is increasing at a constant rate so the v vs t graph will be x (m) (positive/negative/zero) and (increasing/decreasing/constant) t (sec) V vs t The v vs t graph is ____________ at a ________ rate so the a vs t graph will be v (m/s) (positive/negative/zero) and (increasing/decreasing/constant) t (sec) a vs t a (m/s) t (sec) EXAMPLE: X CHANGING MORE QUICKLY AS TIME GOES ON X vs t The x vs t graph is increasing at an increasing rate so the v vs t graph will be x (m) (positive/negative/zero) and (increasing/decreasing/constant) t (sec) V vs t The v vs t graph is increasing at a constant rate so the a vs t graph will be v (m/s) (positive/negative/zero) and (increasing/decreasing/constant) t (sec) a vs t a (m/s2) t (sec) Graphing a free falling body that starts with an upward initial velocity. The slope of the x vs t graph is increasing at a decreasing rate then decreasing and in increasing rate. The graph is steep X vs t with a positive slope at first but ½ way its slope is ____ . After the ½ way point the graph has a __________ slope. This means the v vs t graph starts out positive and decreases at a constant rate and crosses the axis _____________. x (m) t (sec) The slope of the v vs t graph is constant and V vs t (positive/zero/negative). The graph crosses the axis _____________. v (m/s) t (sec) The acceleration is given as the slope of the v vs t graph. Since the slope of the v vs t graph is a vs t (positive/zero/negative) the acceleration is (positive/zero/negative) and constant. a (m/s2) t (sec) What is the velocity of the object when it starts? What is the velocity of the object when it reaches the highest point? What is the acceleration of the object when it starts? What is the acceleration of the object when it is at its highest point? (careful) What is the velocity of the object when it reaches the elevation of the hand on its return fall? .