Mr. G DP Physics Physics and Physical Measurement Topic 1.3 Mathematical and Graphical Techniques

Mr. G DP PhysicsMr. G DP Physics

Physics and Physical Physics and Physical MeasurementMeasurement

Topic 1.3 Topic 1.3

Mathematical and Graphical Mathematical and Graphical TechniquesTechniques


EstimationEstimation•You need to be able to estimate values of everyday objects to one or two You need to be able to estimate values of everyday objects to one or two significant figures significant figures •And/or to the nearest order of magnitude And/or to the nearest order of magnitude •e.g. e.g.

•Dimensions of a brick Dimensions of a brick •Mass of an apple Mass of an apple •Duration of a heartbeat Duration of a heartbeat •Room temperatureRoom temperature •Swimming PoolSwimming Pool

You also need to estimate the result of calculations You also need to estimate the result of calculations

e.g. e.g. 6.3 x 7.6/4.9 6.3 x 7.6/4.9= 6 x 8/5 = 6 x 8/5

= 48/5 = 48/5

=50/5 =50/5

=10 (Actual answer = 9.77)=10 (Actual answer = 9.77)


Approaching and Solving ProblemsApproaching and Solving Problems

• You need to be able to state and You need to be able to state and explain any simplifying assumptions explain any simplifying assumptions that you make solving problemsthat you make solving problems • e.g. Reasonable assumptions as to why e.g. Reasonable assumptions as to why

certain quantities may be neglected or certain quantities may be neglected or ignored ignored • i.e. Heat loss, internal resistance i.e. Heat loss, internal resistance • Or that behaviour is approximately linearOr that behaviour is approximately linear


Graphical TechniquesGraphical Techniques• Graphs are very useful for analysing the data Graphs are very useful for analysing the data

that is collected during investigations that is collected during investigations • Graphing is one of the most valuable tools used Graphing is one of the most valuable tools used

becausebecause• it gives a visual display of the relationship between two it gives a visual display of the relationship between two

or more variables or more variables

• shows which data points do not obey the relationship shows which data points do not obey the relationship

• gives an indication at which point a relationship ceases gives an indication at which point a relationship ceases to be true to be true

• used to determine the constants in an equation relating used to determine the constants in an equation relating two variablestwo variables


• You need to be able to give a You need to be able to give a qualitative physical interpretation of a qualitative physical interpretation of a particular graph particular graph • e.g. as the potential difference increases, e.g. as the potential difference increases,

the ionization current also increases until it the ionization current also increases until it reaches a maximum at…..reaches a maximum at…..

Graphical Techniques cont…Graphical Techniques cont…


Plotting GraphsPlotting Graphs

• Independent variables are plotted on the x-Independent variables are plotted on the x-axis axis

• Dependent variables are plotted on the y-Dependent variables are plotted on the y-axis axis

• Most graphs occur in the 1st quadrant Most graphs occur in the 1st quadrant however some may appear in all 4however some may appear in all 4


Plotting Graphs - Choice of AxPlotting Graphs - Choice of Axiiss

• When you are asked to plot a graph of a When you are asked to plot a graph of a against b, the first variable mentioned is against b, the first variable mentioned is plotted on the y axis plotted on the y axis

• Graphs should be plotted by handGraphs should be plotted by hand


Plotting Graphs - ScalesPlotting Graphs - Scales

• Size of graph should be large, to fill as Size of graph should be large, to fill as much space as possible much space as possible

• choose a convenient scale that is easily choose a convenient scale that is easily subdividedsubdivided


Plotting Graphs - LabelsPlotting Graphs - Labels

• Each axis is labeled with the name and Each axis is labeled with the name and symbol, as well as the relevant unit used symbol, as well as the relevant unit used

• The graph should also be given a The graph should also be given a descriptive titledescriptive title


Plotting Graphs - Line of Best FitPlotting Graphs - Line of Best Fit

• When choosing the line or curve it is best to use a When choosing the line or curve it is best to use a transparent ruler transparent ruler

• Position the ruler until it lies along an ideal line Position the ruler until it lies along an ideal line • The line or curve does not have to pass through The line or curve does not have to pass through

every point every point • Do not assume that all lines should pass through Do not assume that all lines should pass through

the origin the origin • Do not do dot to dot!Do not do dot to dot!


y

x


Analysing the GraphAnalysing the Graph• Often a relationship between variables will first Often a relationship between variables will first

produce a parabola, hyperbole or an exponential produce a parabola, hyperbole or an exponential growth or decay. These can be transformed to a growth or decay. These can be transformed to a straight line relationship straight line relationship

• General equation for a straight line is General equation for a straight line is • y = mx + c y = mx + c

• y is the dependent variable, x is the independent y is the dependent variable, x is the independent variable, m is the gradient and c is the y-variable, m is the gradient and c is the y-interceptintercept

• The parameters of a function can also be obtained The parameters of a function can also be obtained from the slope (from the slope (mm) and the intercept () and the intercept (cc) of a straight ) of a straight line graphline graph


Gradients (Slope)Gradients (Slope)• Gradient = vertical run / horizontal run Gradient = vertical run / horizontal run

• or gradient = or gradient = y / y / x x

• uphill slope is positive and downhill slope uphill slope is positive and downhill slope is negative is negative

• Don´t forget to give the units of the gradientDon´t forget to give the units of the gradient


Areas under GraphsAreas under Graphs

• The area under a graph is a useful tool The area under a graph is a useful tool • e.g. on a force displacement graph the area e.g. on a force displacement graph the area

is work (N x m = J) is work (N x m = J) • e.g. on a speed time graph the area is e.g. on a speed time graph the area is

distance (msdistance (ms-1-1 x s = m) x s = m) • Again, don´t forget the units of the areaAgain, don´t forget the units of the area


Standard Graphs - linear graphsStandard Graphs - linear graphs

A straight line passing through the origin A straight line passing through the origin shows proportionalityshows proportionality

y

x

y x

y = k x

Where k is the constant of proportionality

k = rise/run


Standard Graphs - parabolaStandard Graphs - parabola

A parabola shows that y is directly A parabola shows that y is directly proportional to xproportional to x22

y

x2

y

xi.e. y x2 or y = kx2

where k is the constant of proportionality


Standard Graphs - Standard Graphs - hyperbolahyperbola

A hyperbola shows that y is A hyperbola shows that y is inversely proportional to xinversely proportional to x

y

1/x

y

x

i.e. y 1/x or y = k/x



Standard Graphs - hyperbola againStandard Graphs - hyperbola again

• An inverse square law graph is also a An inverse square law graph is also a hyperbolahyperbola

y

1/x2

y

xi.e. y 1/x2 or y = k/x2



Non-Standard GraphsNon-Standard Graphs

• You need to make a connection between You need to make a connection between graphs and equations graphs and equations

y

x

If this is a graph of r against t2

plotted from data having an expected relationship r = at2/2 +r0 where a is a constant

Then the gradient is a/2 and the y-intercept is r0 - it is not the case that r t2, it is a linear relationship

The intercept is therefore important tooThe intercept is therefore important too

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Mr. G DP Physics Physics and Physical Measurement Topic 1.3 Mathematical and Graphical Techniques