School Science Lessons
(UNPh04)
2024-07-24
Graphs
Contents
Graphs
4.4.1 Linear graphs
4.4.2 Distance / time graph
4.4.3 Cartesian coordinates
4.4.4 Heat one litre of water
4.4.5 Slopes
4.4.0 Graphs
Information that may be obtained from graphs:
* Determine the relationship between two variables and shows the possibility of applying mathematics functions.
* Get the data at any point of the graphs, called interpolation.
It provides the possibility to getting some data not get measured at the experiment.
For example, from graph 4.4.2 you may know how far swimmer A swam in the first 30 seconds.
* Get the data at some point outside the graphs, called extrapolation.
By extending the graphs lines you can get an estimate of data that was not measured.
For example, applying diagram 4.4.2, it may be estimated how far swimmer A would swim after 130 seconds if the swimmer could keep going at the same speed.
However, this would have to be proved by further experiment!
* Get the information about measuring error.
For any graphs drawn carefully, the distribution of separate points at the graphs shows the accidental error of measuring.
The denser the points distribute, the less the accidental error.
* Get other useful information, e.g. maximum and minimum values, the points of intersection between curves and co-ordinate axes, the angles of curves with axes, the area under a curve.
The four main types of graphs
See diagram 4.4.0: Distribution of cats.
A graph is a drawing that shows the relative sizes of quantities or variables.
Graphs show how values change with time or how one variable changes in relation to another.
There are four main types of graph:
* In line graphs or curves, each value is plotted as points on the graph.
The position of each point is given coordinates, which are the distances of the point from two lines called the x-axis (a horizontal line), and the y-axis, a vertical line that crosses the x-axis (at right angles).
* Histograms or bar graphs show information as bars of different heights.
* Picture graphs use small pictures of different sizes or quantities instead of bars.
* Pie charts show the relation of the parts of something to the whole of it, for example, the percentages of a country's budget spent on health, education, defence.
1. Select scales
Select the scale of the axes to make the shape of the graph display the relation between data.
The starting a point of the co-ordinate axis does not have to begin with zero, and the scales of the two axes need not be the same.
The variable you set up is the independent variable and is placed on the horizontal axis, the x axis.
The variable that results from the independent variable is the dependent variable, and is placed on the vertical axis, the y axis.
Show the position of any plotted point as (XY).
If you investigate the cooling of a bucket of water, time is the independent variable and temperature of the water is the dependent variable.
For "Graph speed against time", then "time" is the independent variable, because you have mentioned it after the dependent variable "speed".
2. Plot points and draw by hand
See diagram 4.4.1: Mass and frequency of vibration.
When plotting points in the co-ordinate system by hand the symbols may be small dots surrounded by a circle or a thin cross shape.
When you have two graphs in one co-ordinate system, different symbols should express the points in different graphs.
Do not graph if less than 6 points.
See diagram 4.4.3: Mass and frequency of vibration.
Draw the graph by using the inner drawing method, so that your wrist that is a centre to turn around in forming a smooth graph.
A dotted line should express a graph that you have deduced to distinguish from the graph obtained from experiment.
3. Graph the speed of two cars
See diagram 4.4.2: Speed of two cars.
Suppose you mark a straight road every 10 metres and can use a stopwatch to record when a car reaches each mark.
The following table shows your data for 2 cars, car A and car B.
In the graph the points for Car A are almost in a straight line.
You can say that the line of best fit is a straight line.
The graph line does not go exactly through each point, because some experimental error can occur, reading stopwatch, recording data and plotting the graph.
If you assume that the graph line is properly straight then you can say that each quantity is proportional to the other, distance = speed (velocity) × time, d = vt.
Car A was moving with constant speed 4.2 m / s.
Estimate how far Car A had moved after 8 seconds, by interpolation = 33 m.
See the P on the graph.
Estimate how far Car A had moved after 8 seconds, by calculation, d = vt, d =4.2v × 8 t = 33.6 m.
Car A
Table 3.3.5.0A Graph the speed of two cars A
Distance
(m)		 Elapsed time
(seconds)		 Speed
(m / s)
0 0 0
10 2.3 4.3
20 4.9 4.1
30 7.1 4.2
40 9.7 4.1
50 12.0 4.2
. . Average speed = 4.18 = 4.2
In the graph the points for Car B are not in a straight line.
The line of best fit is a curve so the speed is constantly changing.
The graph shows the method of calculating the instantaneous speed at two distances 15 m and 35 metres.
Draw a tangent to the point on the graph corresponding to the distance.
Construct a right angle triangle with the tangent as hypotenuse. Read the corresponding values for distance and time from the two sides of the triangle then calculate the speed, v = d / t.
At 15 m, the instantaneous speed was s / t, 10 d / 3.2 t = 3.125 m / sec. = 3.2 m / sec.
At 35 m, the instantaneous speed was s / t, 10 d / 7.8 t = 6.25 = 6.2 m / sec.
Car B
Table 3.3.5.0B Graph the speed of two cars B.
Distance
(m)		 Elapsed time
(seconds)		 Instantaneous speed
(m / sec.)
0 0 .
10 4.0 (15 m, 3.2 m / sec
20 7.0 .
30 8.9 (35 m.
6.2 m / sec)
40 10.5 .
50 12.0 .
4.4.1 Linear graphs
See diagram 4.4.8: Linear graph.
A linear graph is a straight line graph which represents a linear relationship between two variables. Linear graphs help us to illustrate how the relationship between these variables changes over time. The simplest relationship between two variables is shown as a straight line graph.
For example, the distance/ time graph of swimmer B in Diagram 4.4.2 is a straight line.
A timer with a stopwatch stands 5m from the pool side where swimmers A and B set out and starts to record the time from zero, when the swimmers pass in front of the timer at nearly the same time.
The relationship of S-T is linear, i.e. y = kx +b, where k is the gradient of the line, b is intercept of the line with y-axis.
Gradient k of the line
See diagram 4.4.8: Gradient k.
The straight line from measuring data is produced by joining points apart from each other.
In reality, measuring data may be not on the actual line, so do not put measuring data into the linear equation directly.
Gradient k may be found by following graphic method.
Suppose A and B are two of any points at the line, C is the point of intersection of a level line through A and a vertical line through B.
The length of AC is equal to the change in x-axis, i.e. (x2 - x1).
The length of BC is equal to the change in y-axis, i.e. (y2 - y1), negative perhaps.
So the gradient of the line k = (y2 - y1) / (x2- x1).
Measure the lengths of AC and BC at the graph.
Gradient k may be calculated by: k = (47 - 19) m / (30 - 10) second, approximately = 1.4 m / s.
When calculating a gradient:
1. Get a triangle from the graph as large as possible, i.e. choose two points farther from each other so that the gradient calculated is more exact.
2. Use the values of two points at the line.
Do not use the two readings at recording, unless the two readings coincide to the line very much.
3. Measure the lengths of relative lines, e.g. AC and BC, through the scale of the axes, but not with a ruler.
4. A gradient usually has its own unit determined by the units of two variables, such as m / s as in the above example.
The unit of a gradient usually shows its meaning in physics.
Intercept of the line b
See diagram 4.4.8: Intercept of the line b.
The distance between the point of intersection of a line with an axis and the origin is called the intercept of the line.
If you can find the gradient k of a line and intercept b of the line with y-axis, the equation of the line may be written.
The equation of the line at diagram 4.4.4.1 is: S = 1.4 t + 5.
The intercept of the line with the x-axis is useful.
For example, extend the line describing the movement of swimmer A at the first lap in the reverse direction.
It intersects x-axis at the point (-12, 0).
Intercept of the line with x-axis a = -12 (s).
It shows how long swimmer A took from starting to swim at some side of the pool to starting to record the time.
The interval is 4 seconds.
Intercept of the line with x-axis is usually expressed as (a = - b / k), i.e. intercept of the line with x-axis is equal to negative ratio of intercept of the line with y-axis to the gradient of the line.
Line graphs
See diagram 4.4.5: Line graphs.
Plot a graph of the measurements of one quantity, y, against the measurements of the other quantity, x.
If the graph is a straight line passing through the origin there is a simple relationship between two sets of measured quantities.
If a straight line passes through the origin (0, 0), the constant gradient k = y / x, so y = kx.
See diagram Y = X2.
4.4.2 Distance / time graph
See diagram 4.4.7: Distance / time graph of two swimmers.
The graphs of distance against time in diagram 4.4.2 describes the distance travelled and time taken by two swimmers in a race of two laps of a 50 metre pool.
From qualitative analysis of the graphs you can say:
1. After 25 metres of the first lap, Swimmer A was faster than Swimmer B (the graph line of Swimmer A is steeper than the graph line of swimmer B).
2. After 50 metres, the end of the first lap, Swimmer A was faster than Swimmer B (the graph line of Swimmer A is steeper than the graph line of swimmer B).
3. During the front 25m of the second lap, they kept the same distance apart (the graph lines are parallel.).
4. After 50 metres of the second lap, the end of the race, Swimmer B was faster than swimmer A (the graph line of B is much steeper than the graph line of A).
5. Swimmer A preceded swimmer B for a longer period, but finally Swimmer B won.
6. The shape of the curve shows why swimmer A failed to win.
Swimmer A swam at the second lap more slowly than that at the first lap.
By contrast swimmer B swam at the second lap faster than that at the first lap and especially during the final 1 / 4 distance of the second lap swimmer B swam very fast.
4.4.3 Cartesian coordinates
See diagram 4.4.9: Intercept of the line b.
4.4.4 Heat one litre of water
Heat one litre of water from room temperature to 100oC.
Before the experiment, set up a tripod, a heating mat, and place the beaker on the mat.
Design a table of results.
Hold the thermometer vertically and slowly put it into the water until the liquid bulb of the thermometer immerses into the water completely.
Do not stir water with the thermometer.
The thermometer must remain in the water while observing and should not touch the bottom or wall of the beaker.
Then read the value of the temperature of water with eyes being the same level of the liquid column.
The temperature of the water should the same as the room temperature.
Light the Bunsen burner and turn the sleeve around to get a non-luminous flame.
Slide the Bunsen burner under the beaker to heat the beaker evenly.
Keep stirring and steadily heat the water while recording the temperature every 30 seconds.
Draw a graph of variation of temperature in water with the time.
Plot temperature on the vertical axis and time on the horizontal axis.
Connect each data point to get a smooth graph.
Reach a conclusion according to the shape of the graph.
Repeat the experiment with a fan blowing slowly on the apparatus.
The temperature rises rapidly at first heating, because the rate of evaporation is slow at that temperature and loss of heat to the surroundings is slow as long as the temperature difference between the water and room temperature is small.
As the temperature of the water rises, the rate of rise of temperature diminishes at an increasingly rapid rate.
4.4.5 Slopes
See diagram 4.4.10: Slopes.