Graphing Motion Vocabulary: The Slope is the Thing trend slope “Composition with Red Blue and Yellow” (1930) Piet Mondrian This topic can be found in your textbook on pp. 45-46 and 50-52.

What is the purpose of a graph? To give a visual representation of the relationship between variables.

Atmospheric Concentrations of CO2 In physics (and in science in general), we are very interested in trends. What is going on in this graph? Atmospheric Concentrations of CO2 x-variable? y-variable? Trend? Any ideas about blue vs. red? What does the slope of the blue line on this graph tell you?

Atmospheric Concentrations of CO2 Slope = Δy/Δx. If x is time, then slope tells you the rate of change of your y-variable. Atmospheric Concentrations of CO2 On this graph, the slope tells you the rate of change of atmospheric carbon dioxide. Average slope = average rate of change of CO2 Slope at a point = instantaneous rate of change of CO2 Was the rate of CO2 concentration increasing faster in 1965 or 1995? Note that this is a different question than “was the concentration higher in 1965 than in 1995?”

Example Does point A or point B have a higher share value? Price ($) Does point A or point B have a higher share value? Looking at the y-axis, B has the higher share value Is the rate of change of the share value greater at A or B? Looking at the slopes, A has the greater rate of change B A

Graphing Velocity Remember that slope is the ratio of the y- variable and the x-variable. 𝑚= ∆𝑦 ∆𝑥 Also, remember that velocity is the ratio of displacement and time change. 𝑣= ∆𝑥 ∆𝑡 So if we graph displacement (Δx) on the y-axis, and time (Δt) on the x-axis, the slope will be velocity (the rate of change of position)!

Example The graph below shows the motion of Wanda Wren in her airplane. Has Wanda travelled farther at 0.10 s or at 0.40 s? 0.40 s, because she has a greater change in position at 0.40 s

Example Does Wanda have a greater velocity at 0.10 s or at 0.40 s? 0.40 s, because the slope is greater at 0.40 s

Important Point The slope on a position-time graph (position on the y-axis and time on the x-axis) is equal to the velocity. Average slope is average velocity and the slope at a point is instantaneous velocity.

Practice: The graph shows the motion of Howie Doohan mowing his lawn Practice: The graph shows the motion of Howie Doohan mowing his lawn. Qualitatively describe the motion of Howie. From t = 0s to t = 5s, Howie is moving forward with constant velocity (positive constant slope). From t = 5s to t = 10s, Howie is stationary (zero slope). From t = 10s to t = 12.5s, Howie is moving backward with constant velocity faster than he had moved forward (steeper, negative constant slope).

Practice Important points + slope = + v (moving forward) - slope = - v (moving backward) 0 slope = stationary steeper slope = faster speed

Practice Quantitative description of motion What is Howie’s velocity from t = 0s to t = 5s? 𝑣=𝑠𝑙𝑜𝑝𝑒 = ∆𝑥 ∆𝑡 = 𝑥 2 − 𝑥 1 𝑡 2 − 𝑡 1 = 5𝑚 −0𝑚 5𝑠 −0𝑠 =1 𝑚 𝑠

Practice Quantitative description of motion What is Howie’s velocity from t = 5s to t = 10s? 𝑣=𝑠𝑙𝑜𝑝𝑒 = ∆𝑥 ∆𝑡 = 𝑥 2 − 𝑥 1 𝑡 2 − 𝑡 1 = 5𝑚 −5𝑚 10𝑠 −5𝑠 =0 𝑚 𝑠

Practice Quantitative description of motion What is Howie’s velocity from t = 10s to t = 12.5s? 𝑣=𝑠𝑙𝑜𝑝𝑒 = ∆𝑥 ∆𝑡 = 𝑥 2 − 𝑥 1 𝑡 2 − 𝑡 1 = 0𝑚 −5𝑚 12.5𝑠 −10𝑠 =−2 𝑚 𝑠

Qualitative Practice Where is L. Paso stationary? The following graph shows L. Paso’s motion while riding his horse. Where is L. Paso stationary? Answer: Between C and D

Qualitative Practice Where is L. Paso moving forward with constant velocity? Answer: Between B and C

Qualitative Practice Where does L. Paso have a constant acceleration (not = 0)? Answer: Between A and B

Important Point Curves on position-time graphs show acceleration.

Qualitative Practice Where does L.’s acceleration = zero? Answer: Between B and C, between C and D, and between D and E (but not including point B, C, D, and E…why not?)

Qualitative Practice Where is L. Paso moving backward? Answer: Between D and E

Qualitative Practice Ignoring A to B, Where does L. Paso have the greatest speed? Answer: Between D and E

Next topic: Graphing Acceleration Remember that slope is the ratio of the y- variable and the x-variable. 𝑚= ∆𝑦 ∆𝑥 Also, remember that acceleration is the ratio of velocity and time change. 𝑎= ∆𝑣 ∆𝑡 So if we graph velocity (Δv) on the y-axis, and time (Δt) on the x-axis, the slope will be acceleration(the rate of change of velocity)!

Example The graph below shows the motion of I.M. Fast in his car. Is I.M. speeding up or slowing down? How do you know? Is I.M.’s accleration changing or constant? How can we calculate I.M.’s acceleration? Calculate the slope!

Example 𝑎=𝑠𝑙𝑜𝑝𝑒 = ∆𝑣 ∆𝑡 = 𝑣 2 − 𝑣 1 𝑡 2 − 𝑡 1 = ∆𝑣 ∆𝑡 = 𝑣 2 − 𝑣 1 𝑡 2 − 𝑡 1 = 16𝑚 − 4𝑚 4𝑠 −1𝑠 =4 𝑚 𝑠 2 What does this value of acceleration mean? It means I.M. is speeding up by 4 m/s per s. You can clearly see this on the graph by looking at I.M’s velocity every second of his travel time.

Example 𝑎=4 𝑚 𝑠 2 What would a graph of the acceleration vs. time look like for this motion? Since it is a constant acceleration, it will be a straight line with zero slope at a = 4 m/s2. Acceleration (m/s2)

Example The following graph shows Pearl E Example The following graph shows Pearl E. Gates’ 2 stage model rocket flight.

Example Where on the graph is Pearl’s rocket moving downward? Answer: Between 9s and 14s.

Example Where on the graph is Pearl’s rocket slowing down? Answer: Between 4s and 9s.

Example Describe the difference between the motions when Pearl’s rocket has zero velocity at t = 0s and at t = 9 s. Answer: At t = 0s, the rocket is just starting and speeding up in positive direction. At t=9s, it has just changed directions and is speeding up in the negative direction.

Example Calculate the two different accelerations between t = 0s and t = 4.0 s. Answer: 40 m/s2 and 20 m/s2

Example: Make a position-time graph and an acceleration-time graph of the following motion First, let’s identify places where the motion changes. This includes changes in slope or changes in direction of movement. Red: accelerating in the positive direction (positive acceleration) Orange: moving with constant velocity in the positive direction (zero acceleration) Green: slowing down and moving in the positive direction (negative acceleration) Blue: speeding up in the negative direction (negative acceleration) v (m/s) t (s)

Example: Make a position-time graph and an acceleration-time graph of the following motion Now let’s draw each motion independently so we can focus. Let’s make the position-time graph first. v (m/s) x (m) t (s) t (s)

Now let’s draw the acceleration-time graph. Example: Make a position-time graph and an acceleration-time graph of the following motion Now let’s draw the acceleration-time graph. v (m/s) x (m) t (s) t (s)

Graphing Summary The slope of a position time graph is velocity. The slope of a velocity time graph is acceleration. 𝑠𝑙𝑜𝑝𝑒= ∆𝑦 ∆𝑥 = 𝑦 2 − 𝑦 1 𝑥 2 − 𝑥 1