graphs

April 2000 Graphs - 1 Physics@Xinmin

Contents1. Reading Graphs2. Gradient3. Plotting Graphs

1. Reading GraphsGraphs in Physics will have lines that are either straight-lines or smooth-curves.Graphs may, however, consist of several of these lines.

Example of a straight-line graph Example of a smooth-curve graph

Some will pass through the origin (0,0), some will not.


Example:Use the graph, showing the speed of a car against time, to answer the questions

1. What are the ranges of the x-axis and the y-axis?

2. What is the speed of the car after 1.2 seconds?

3. How long does it take for the car to reach a speed of 7 m/s?

4. At which two instants (times) does the car move with a speed of 15 m/s?

5. What is the maximum speed with which the car moves?

6. At what time does the car travel at its maximum speed?

7. Can the graph tell us whether the car is going uphill or downhill?

8. Can the graph tell us what colour the car is?


2. GradientLook at the two graphs below.

Q. What is the difference between them?

Calculating the Gradient of a Straight-Line GraphWe can measure the slope (gradient) of the line and express it as a number.

1. Choose two points (P1 and P2) one near to each end of the graph.

P2 (x2,y2)

P1 (x1,y1)

2. Mark out a large triangle using the two points you have chosen.

P2 (x2,y2)

P1 (x1,y1)

3. To find the gradient we use the formula

Gradient = Increase in y valuesIncrease in x values

m = y2 – y1x2 – x1


Worked example:

0

5

10

15

20

25

30

35

40

45

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

105


Exercise:Find gradients of the following lines:Q1.

0123456789

101112131415161718

0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425

Q2.

0123456789

101112131415161718

0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425

Q3.

0123456789

101112131415161718

0 1 2 3 4 5 6 7 8 9 10111213141516171819202122232425


Refer to the following graph for Q4.-Q6.

Q4. Which of the lines has the largest gradient?

Q5. Which of the lines has a negative gradient?

Q6. Which of the lines has a gradient of zero?

Calculating the Gradient of a Smooth-Curve Graph

1. Choose the point at which you wish to find the gradient.

2. Draw a tangent to the line at the chosen point.

3. Draw a triangle using the tangent and calculate the gradient.

Look at the graph below. Mark on the diagram the point, G, where the gradient is thegreatest.

A

BC

D


3. Plotting GraphsGraphs are plotted to show the relationship between two quantities. The quantity thatyou control and change is usually plotted on the x-axis. You should vary this quantityin regular steps in the experiment. The quantity that is dependent on the quantity thatyou control or change is plotted on the y-axis.

The following points should be noted when drawing graphs.

a) Label both axes prominently with the names and units of the variables. The SImethod is recommended e.g. “x/m” or “I/A” etc.

b) Give a title to the graph.c) Use a convenient scale to draw the graph as large as available space allows.

(i) Avoid using “3-scales” and other awkward scales. Such scales usuallylead to errors in plotting and reading from the graph.

(ii) The paper can be placed either in the landscape or portrait position.

d) Do not attempt to join all of the points on the graph. It is not likely that youwould obtain a straight line or a smooth curve. Rather, use a transparent ruler tohelp you draw the best straight line or a flexi-curve to draw a smooth curvethrough most of the points.

e) When determining the gradient of a straight line, draw a large triangle and usethe co-ordinate method to determine the gradient.

f) Evidence of how a reading is obtained from the graph must be shown.(i) When finding the gradient ensure that a triangle is drawn on the graph

paper.(ii) Reference lines to find a point on the horizontal axis corresponding to a

point on the vertical axis or vice-versa.

The following two diagrams show good examples of straight-line and smooth-curvegraphs.

Notice how the lines do not pass through every point but pass close to all of the points.


Exercise:Look at the following graphs and determine if they are drawn correctly or if there areany mistakes. For the cases where there are mistakes state the mistake and suggest away that they could be corrected. 1.

2.

3.

4.


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