Page 1 Graphical Analysis of Motion By: Mike Maloney

Mike Maloney © 2004Page 2 Graphing Motion A common way in physics to describe different phenomena is with graphs. Scientists often take data, plot the data, and they analyze the shape of the graph to try to determine relationships between things. Our first goal this year will be to identify, describe, and analyze motion using graphical techniques.

Mike Maloney © 2004Page 3 Position vs. Time (s vs. t) One way to graphically describe the motion of an object is to plot a graph of its Position vs. Time. It is common practice to put Time on the horizontal-axis of a graph. So we will put Position on the vertical-axis. A position vs. time graph may look something like this …

Mike Maloney © 2004Page 4 s vs. t

Mike Maloney © 2004Page 5 s vs. t From this graph we can tell a few things about the motion of our object. First we can tell the position of the object at any point in time by reading it off the graph. If the graph increases, the object moves in the (+) direction. If the graph decreases, the object moves in the (-) direction. What does it mean if the graph goes into the negatives? –The position is negative, or you might say you are behind what you called the “0” position.

Mike Maloney © 2004Page 6 Interpreting an “s vs. t” Now let’s can see what “s vs t” plots tell us about motion.

Mike Maloney © 2004Page 7 Interpreting an “s vs. t” We know that we can read the position of an object directly from a “s vs t” plot.

Mike Maloney © 2004Page 8 Interpreting an “s vs. t” At 1.5 seconds, where is the object? Between 8 and 9 meters.

Mike Maloney © 2004Page 9 Interpreting an “s vs. t” We can also determine the change in position by reading values from the graph.

Mike Maloney © 2004Page 10 Interpreting an “s vs. t” How far does it move from 1 to 3 seconds?

Mike Maloney © 2004Page 11 Interpreting an “s vs. t” How far does it move from 1 to 3 seconds?  s = 14 – 5 = 9 meters. 9

Mike Maloney © 2004Page 12 Slope of “s vs. t” Now let’s try do determine some not so easy to see data from our “s vs t” plot.

Mike Maloney © 2004Page 13 Slope of “s vs. t” How do you find the slope of a line? change in y / change in x

Mike Maloney © 2004Page 14 “s vs. t” So what would the slope of a “s vs t” plot give us? change in y / change in x change in s / change in t or  s /  t the rate of change of position, or the velocity.

Mike Maloney © 2004Page 15 So far … (s vs. t) A position vs. time plot tells us the position (s) of an object at any point in time It can either be positive or negative Positive (+): positive of where you started Negative (-): negative of where you started The change in “y” on a position vs. time plot from one point to another gives us the displacement (  s) Going Up (+change): moving in the positive direction Going Down (-change): moving in the negative direction The slope of the position vs. time plot tells us the velocity

Mike Maloney © 2004Page 16 Velocity vs. Time (v vs. t) Another way to graphically describe motion is to make a plot of the velocity of an object versus time. Again we will put Time on the horizontal-axis, and Velocity on the vertical-axis A plot of a velocity vs. time graph may look like this …

Mike Maloney © 2004Page 17 v vs. t

Mike Maloney © 2004Page 18 v vs. t From this graph we can tell the velocity of an object at any point in time.

Mike Maloney © 2004Page 19 v vs. t Notice that this graph has points both positive and negative.

Mike Maloney © 2004Page 20 v vs. t What does the positive part of the graph indicate? The times when the velocity was positive, or when the object was moving in the positive direction.

Mike Maloney © 2004Page 21 v vs. t What about the negative part of the plot? The times when the velocity was negative, or the object was moving in the negative direction.

Mike Maloney © 2004Page 22 v vs. t As the plot increases in the positive direction, the speed of the object increases.

Mike Maloney © 2004Page 23 v vs. t As it goes down in the positive part of the graph, the speed of the object decreases.

Mike Maloney © 2004Page 24 v vs. t As the plot goes down in the negative section, the speed also increases, but in a negative direction.

Mike Maloney © 2004Page 25 v vs. t This would be symbolic of increasing a car’s speed in reverse.

Mike Maloney © 2004Page 26 v vs. t As the plot goes up in the negative section, the speed decreases in the negative direction.

Mike Maloney © 2004Page 27 v vs. t This would be symbolic of slowing down in reverse.

Mike Maloney © 2004Page 28 Speed and Velocity This brings up a common confusion in physics … why do we have speed and velocity? Speed is just the number part, or the size of the velocity, it is what the speedometer in you car reads. Velocity not only tells you how fast, but also in what direction you are going.

Mike Maloney © 2004Page 29 So far … (v vs. t) A velocity vs. time plot tells us the velocity of an object at any point in time It can either be positive or negative Positive: (+) velocity [+direction] Negative: (-) velocity [-direction] Going Up (+part): increasing positive speed, velocity goes up Going Down (+part): decreasing positive speed, velocity goes down Going Up (-part): decreasing negative speed, velocity goes up Going Down (-part): increasing negative speed, velocity goes down

Mike Maloney © 2004Page 30 v vs. t So we can read the velocity of an object directly off the chart.

Mike Maloney © 2004Page 31 Like in other graphs, we can also talk about its slope. What is the slope of a “v vs. t” graph? v vs. t

Mike Maloney © 2004Page 32 Slope = change in y / change in x = change in velocity / change in time Slope =  v /  t = acceleration v vs. t

Mike Maloney © 2004Page 33 The area underneath the curve gives us something also. How would we find the area underneath the curve from 1 sec to 3 sec? v vs. t

Mike Maloney © 2004Page 34 Area = height x base = velocity x change in time Area = v *  t that is the equation for displacement (  s). v vs. t

Mike Maloney © 2004Page 35 a vs. t We can extrapolate our ideas on position and velocity graphs to acceleration vs. time plots. The Area under the curve in an a-t graph is –(Acceleration) · (time) –a ·  t –Which is change in velocity (  v). The slope (  a/  t) actually gives us something called the jerk, but we don’t have to worry about that in this class.

Mike Maloney © 2004Page 36 In Conclusion Slope of “s vs. t” plot is velocity Slope of “v vs. t” plot is acceleration Area under “v vs. t” plot is displacement Area under “a vs. t” plot is  velocity Now lets try some …