graphing#from#yesterday’s#lab.# Remember,#abestﬁtline#should#not“connect the#dots… · •...

•  10 minutes to finish any data analysis or graphing from yesterday’s lab.

•  Remember, a best fit line should not “connect the dots”, in this case it should be a straight line that contains as many data points as possible.

Using pictures to represent mo?on

•  The easiest way to represent mo?on through using an image is the par?cle diagram.

•  We imagine that the moving object is dropping something each second.

•  We use that dropped object to keep track of the type of mo?on

•  Quick Cheats •  Evenly-‐spaced par?cles mean a constant speed. Why? – Because constant speed means you cover the same distance each second.

•  Par?cles that are geMng further apart mean accelera?on. Why?

•  So what do par?cles geMng closer together mean?

•  Limited use – Can’t tell us direc?on of travel – No good if object changes direc?on

•  Instead we’ll focus most of our ?me on using x/y graphs of mo?on – But first, we need a refresher on the types of graphical rela?onships out there.

•  straight line graph •  the variables are directly propor?onal to each other

–  As x increases, the value of y increases by a set amount

Our standard linear equa?on is y = mx + b m = slope, b = y-‐intercept

•  Each graph is based on a par?cular equa?on. – Rearrange equa?on – Check if variables fit the linear model

•  EXAMPLE ON WHITEBOARD – Copy this example into your notes

•  Finding slope is just division – SLOPE IS SUPER IMPORTANT IN THIS CLASS

•  Interpre?ng slope: what quan?ty does the slope of a graph represent? – Analyze the units used to make the data points

•  LETS GO BACK TO THAT WHITEBOARD EXAMPLE (COPY THIS TOO)

•  Exponen?al : y = x2

•  Inverse: y = 1/x

•  Inverse exponen?al: y = 1/x2

•  y = x2 •  As x increases, y increases at a squared rate. – X = 1, y =1: x = 2; y = 4; x = 5, y = 25

This graph shows half of a parabola. We can qualita?vely analyze slope, but not find the actual numbers.

•  y = 1/x •  As x increases, y decreases – x = 2, y = ½; x = 4, y = ¼; x = 1000, y = 1/1000

This graph shows a curve with a decreasing slope as the y-‐value approaches zero.

This looks very similar to our regular inverse graph only it drops to zero much more quickly. Why does this and the inverse graph blow up when x<1? So as x increases, y decreases….a lot.

•  For the following equa?ons, figure out whether graphing the listed quan??es would result in a linear, exponen?al, or inverse shape.

•  Eg: F = ma : y = F, x = a – We expect this graph to be linear because neither quan?ty is squared and a is in the numerator.

•  P = W/t: y = P, x = t •  Fe = (kq1q2)/r2 : y = Fe , x = r •  Wt = VI: y =W, x = t •  d = vit + ½ at2: y = d, x = t

•  Graphs which represent the mo?on of objects •  Mo?on graphs can tell us speed, direc?on, and how these change over ?me

•  We will analyze two types of mo?on graphs – Distance/displacement vs ?me – Speed/velocity vs ?me

Linear graph This tells us that there’s no accelera?on. Why? How could we find the average speed by using this graph?

Sketch the shape of the graph below (it doesn’t need to be perfect) and answer the following:

-‐ What is the speed of the object from 5-‐15s? -‐ Does the object ever change direc?on? -‐ Is it going faster the first five seconds or the last five?

•  Cut and paste the graphs into the len side of your notebook and write a descrip?on of the mo?on shown in the graph.

5:00

Obj: Swbat analyze velocity vs ?me graphs in order to find accelera?on.

•  Cut and paste the graphs into the right side of your notebook and write a descrip?on of the mo?on shown in the graph.

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•  How do we know that d = vt will give us a linear graph?

•  What type of d vs t graph do we expect when we have a moving object with posi?ve accelera?on? Why do we get this shape?

•  What does the slope of a d vs t graph tell us? How do we figure out the meaning of the slope for any graph?

•  Creates a curved line on any d vs t graph •  If a>0 (posi?ve), the slope of the curve is increasing •  If a<0 (nega?ve), the slope of the curve is decreasing

–  This means the curve is becoming a horizontal line

Posi?ve accelera?on nega?ve accelera?on

•  Seven regents problems to help cement your knowledge of d vs t graphs.

•  Work on your own for 10 minutes (silent) •  Once the ?mer gets to 5min, you may work together.

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Time’s up!

Let’s go over the answers

•  Final 100 point grade on Friday

•  Fix your table of contents •  Pay aven?on to the rubric •  If you’re missing do nows, you’ll want to find them

•  Remember, being absent is NOT an excuse. It’s your responsibility to get this info

•  GO TO PHET! NO MORE NOTESSSSSS

1. Below is a graph of a basketball’s mo?on. Which of the following gives the best interpreta?on of the ball’s mo?on?

a. The ball moves along a flat surface. Then it moves forward down a hill, and then finally stops.

b. The ball doesn’t move at first. Then it moves forward down a hill and finally stops.

c. The ball is moving at constant velocity. Then it slows down and stops.

d. The ball doesn’t move at first. Then it moves backwards and then finally stops.

e. The ball moves along a flat area, moves backwards down a hill and then it keeps moving.

Draw the graph and write the descrip?on of your answer

•  d vs t graphs are actually Δd vs t –  Graph shows us how displacement changes over ?me

•  In order to graph Δv, what equa?on would be easiest to use? –  a = Δv/t : Now we need to rearrange the equa?on so that the quan?ty we’re graphing is alone

–  Δv = at – What kind of graphical rela?onship do we expect from this equa?on?

•  In this class, we will only look at the v vs t graph of this equa?on.

•  Our graph is based on the equa?on Δv = at –  If we’re comparing this to y = mx + b, which quan?ty is our slope?

•  We can analyze slope by using the units of our y/x quan??es –  Δy/Δx = Δv/Δt –  v [m/s] , t [s] –  So what is (m/s)/s?

•  Using our slope equa?on we see that the units of slope for our v vs t graph are [m/s2]

Sketch this graph in your notebook and find the accelera?on for each sec?on.

On len side: Sketch graph and for each sec?on describe the mo?on. Which way is the object moving? Speeding up or slowing down? Use East as posi?ve

•  Two ways to find displacement from a v vs t graph –  Use the graph to construct kinema?cs problem –  Find the area under the graph.

•  Example: copy the figure into your notes and the work from the board

•  Sketch the graph and calculate the total displacement of the object depicted

•  You have 22 min to complete this work – 10 min solo and silent – 12 minutes with group

•  Use your notes to help you figure out what’s going on.

•  Take a d vs t graph and translate it into a v vs t – Find the velocity of each sec?on (?me) – Graph that onto a v vs t graph

•  Mostly we just want to match general shapes

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