instantaneous velocity calculator
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now once again were going to look at the graph of an objects position over time and we can see in this graph how the objects position changes over time and were going to talk about the average velocity and the instantaneous velocity and the average velocity will be a measure of how much how much distance the object covers over a time interval and the instantaneous velocity will be a measure of how fast the object is moving at an instant at a particular moment in time so lets look at the average velocity first lets put some numbers on here suppose this point represents a time of two seconds and the objects at a position of 20 meters and then lets suppose over here lets suppose this point represents a time of 12 seconds and corresponds to a position on the x-axis of 40 meters now we can connect those two points we can draw a line segment connecting those two points the slope of that line segment will be the average velocity over that time interval you should be able to see that between two seconds and 12 seconds during that time interval it went from 20 meters to 40 meters so we can talk about this change in X and this change in time and the average velocity lets calculate that over here the average velocity is going to be Delta x over delta T the change in position compared to the corresponding time interval so thats going to be 20 meters you can see on the graph the change in position going from 20 to 40 is a change of 20 meters and the time interval is 10 seconds so 20 meters over 10 seconds comes out to 2 meters per second thats the average velocity over that time interval now part of the time it was moving forward part of the time it was moving backward and part of the time it was moving forward again but on average it was moving 2 meters per second forward and we can write this we can write that the slope of a segment connecting two points on an XT and thats this segment that I drew in here and Im going to write this you should write this in your notes too the slope of a segment connecting two points on an xt graph the slope of a segment connecting two points on an xt graph is the average velocity over that time interval the slope of the segment connecting two points on an xt graph is the average velocity over that time interval now you can memorize that fact but your goal should be to understand that fact think about it it should make sense you connect any two points on an xt graph you get a line segment the slope of that line segment is the average velocity over that corresponding time interval sometimes though we dont want to find the average velocity we want to find a velocity at a particular moment so for example what if we had this moment in time right here five seconds and that corresponds to this point on the graph and we want to know how fast its moving at that particular instant one moment in time so theres no no time interval here instead we have an instant one particular time and we want to know the velocity at that particular moment we need the slope at that point we need the slope of the graph at that point so this
is this corresponds to a particular x value here on the graph but if we try to calculate the slope here we cant do a rise over run we cant do a delta x over a delta T because this is just one point right here and so is this this is just one point so our Delta X here would be infinitely small and our delta T would be infinitely small if were trying to calculate a slope of an infinitely small point on the graph right there you can see the graph does have a particular steepness the graph right here is steeper than it is say here or here but we cant do rise over run because were talking about one little point Ive drawn a little line segment but we have to imagine that little line segment being infinitely small now we could imagine this graph being made up of a bunch of little tiny straight line segments and at a particular moment in time one of those segments would have a particular steepness but to be accurate we would have to imagine an infinite number of infinitely small little line segments the slope of each one would be the slope of the graph at a particular point and that slope that were trying to find right here in this case is what we call the derivative the word derivative really just means the same thing as slope and we recognize that it can be different at every point on the graph and when we have when we have said already that the slope of an X versus T graph is the velocity another way to say that is to say that the derivative of the position is the velocity because derivative essentially means the same thing as the slope now to find the slope at this particular point heres what were going to do were going to draw a line tangent to the graph at this point so you want to draw a line that just barely touches the graph at that one point and if you do it right this line well have the same steepness as this curve has at that point and we say that this line this line is called a tangent line it is tangent to the graph at that point and the slope of the line tangent to the graph at that point is equal to the slope of the graph at that point and we can calculate the slope of that line segment we can just pick two points on the line segment any two points and these two points would correspond to an interval on the x-axis and an interval on the T axis and with that we can calculate a slope of that line segment you can see we now have a discernible rise and a discernable run and that would be in meters and in seconds so we would get a slope in meters per second in this case but instead of having Delta X and delta T being infinitely small now we have an actual number for Delta X and an actual number for delta T that we can use to calculate a velocity and that would be the slope represented by this line and that would be the velocity at that moment at that instant what we call the instantaneous velocity and we would also say that the value the number we get for the slope of this segment is the derivative of this graph at that point
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