Why Distance is Area under Velocity-Time Line

<?xml version="1.0" encoding="UTF-8"?>  <XholonWorkbook> <Notes> Khan Academy ------------ Title: Why Distance is Area under Velocity-Time Line Description: Understanding why distance is area under velocity-time line Url: http://www.khanacademy.org/video/why-distance-is-area-under-velocity-time-line?playlist=Physics InternalName: why-distance-is-area-under-velocity-time-line YoutubeId: d-_eqgj5-K8 Keywords: physics, area, integral, accleration, slope My Notes -------- An object is moving at 5 m/s to the right. velocity v is a vector displacement s is a vector Sal draws a graph 0:20 the velocity is constant How far does it travel after 5 seconds? v = s / delta t v * delta t = s 5 m/s * 5 s = 25 m this is exactly the area under the velocity curve in the graph 2:15 In general, if you plot the magnitude of the velocity, the area under the curve will be the displacement. An example 2:50 acceleration = 1 m/s^2 Sal draws another graph magnitude of initial velocity (initial speed) = 0 acceleration is the slope of the curve a = delta v / delta t the slope is constant because the acceleration is constant How far does the object travel if it moves at constant acceleration for 5 seconds? 5:27 vi = 0, a is constant Sal draws increasingly tiny rectangles under the curve (a diagonal line) Can calculate the answer as the area of a triangle = 0.5 * base * height displacement s = 0.5 * 5 seconds * 5 m/s = 12.5 meters x axis is time y axis is magnitude of the velocity slope is the acceleration area under the curve is the displacement For this example, I will draw a chart using google or JFreeChart. </Notes> <script implName="lang:python:inline:"><![CDATA[ dt = 5.0 # delta t in seconds (base of a triangle) dv = 5.0 # delta v in m/s (height of a triangle) s = 0.5 * dt * dv # displacement in meters print "displacement = " + str(s) + " meters" # result: displacement = 12.5 meters ]]></script> <script implName="lang:javascript:inline:"><![CDATA[ dt = 5.0; // delta t in seconds (base of a triangle) dv = 5.0; // delta v in m/s (height of a triangle) s = 0.5 * dt * dv; // displacement in meters println("displacement = " + s + " meters"); //javascript:(function(){dt=5;dv=5;s=0.5 * dt * dv;alert("displacement = "+s+" meters");})() // result: displacement = 12.5 meters ]]></script> <_-.XholonClass>  <PhysicalSystem/> <Block/>   <Time superClass="Duration"/> <Displacement superClass="Length"/> </_-.XholonClass> <xholonClassDetails> <Block> <port name="acceleration" connector="#xpointer(Acceleration)"/> <port name="time" connector="#xpointer(Time)"/> <port name="velocity" connector="#xpointer(Velocity)"/> <port name="displacement" connector="#xpointer(Displacement)"/> </Block>  <Velocity xhType="XhtypePurePassiveObject"/> </xholonClassDetails> <PhysicalSystem> <Block> <Acceleration>1.0 m/s^2</Acceleration> <Time>0.0 s</Time> <Velocity>0.0 m/s</Velocity> <Displacement>0.0 m</Displacement> <Blockbehavior implName="org.primordion.script.Behavior_javascript"><![CDATA[ var behaviorObject = new Object(); behaviorObject.postConfigure = function() { var app = applicationKey; app.setUseDataPlotter('JFreeChart'); // google JFreeChart app.setDataPlotterParams('Graph of an object moving at constant acceleration,Time (s),||Velocity|| (m/s),./statistics/,stats,1,WRITE_AS_LONG'); app.createChart(contextNodeKey.getParentNode()); println(time.val + ',' + velocity.val + ',' + displacement.val); } behaviorObject.act = function() { var dt = 1.0; time.incVal(dt); var dv = acceleration.val * dt; // displacement is the total area under the curve // = current displacement + a rectangular area due to velocity + a triangular area due to acceleration displacement.setVal(displacement.val + (velocity.val * dt) + (dv * dt * 0.5)); velocity.incVal(dv); println(time.val + ',' + velocity.val + ',' + displacement.val); }]]></Blockbehavior> </Block> </PhysicalSystem> <Blockbehavior implName="lang:python:inline:"><![CDATA[ # TODO ]]></Blockbehavior> <Blockbehavior implName="lang:javascript:inline:"><![CDATA[ // TODO ? ]]></Blockbehavior> <Blockbehavior implName="lang:bsh:inline:"><![CDATA[ // This works if pasted in as a last child of Block. //height.incVal(0.02); //System.out.print("Java/Beanshell wants something to do. Height:" + height + "\n"); ]]></Blockbehavior> <Blockbehavior implName="lang:jruby:inline:"><![CDATA[ #require 'java' # This works if pasted in as a last child of Block. #$height.incVal(0.02) #puts "Ruby wants something to do. Height: #{$height}" ]]></Blockbehavior> <Blockbehavior implName="lang:groovy:inline:"><![CDATA[ // This works if pasted in as a last child of Block. //height.incVal(0.02); //System.out.print("Groovy wants something to do. Height:" + height + "\n"); ]]></Blockbehavior> </XholonWorkbook>