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

2. Mr. G DP Physics Estimation You need to be able to estimate values of everyday objects to one or two significant figures You need to be able to estimate values of everyday objects to one or two 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 brickDimensions of a brick Mass of an appleMass of an apple Duration of a heartbeatDuration of a heartbeat Room temperatureRoom temperature Swimming PoolSwimming Pool  You also need to estimate the result of calculations e.g. 6.3 x 7.6/4.9 = 6 x 8/5 = 48/5 =50/5 =10 (Actual answer = 9.77)

3. Mr. G DP Physics Approaching and Solving Problems You need to be able to state and explain any simplifying assumptions that you make solving problems You need to be able to state and explain any simplifying assumptions that you make solving problems e.g. Reasonable assumptions as to why certain quantities may be neglected or ignored e.g. Reasonable assumptions as to why certain quantities may be neglected or ignored i.e. Heat loss, internal resistance i.e. Heat loss, internal resistance Or that behaviour is approximately linear Or that behaviour is approximately linear

4. Mr. G DP Physics Graphical Techniques Graphs are very useful for analysing the data that is collected during investigations Graphs are very useful for analysing the data that is collected during investigations Graphing is one of the most valuable tools used because Graphing is one of the most valuable tools used because it gives a visual display of the relationship between two or more variablesit gives a visual display of the relationship between two or more variables shows which data points do not obey the relationshipshows which data points do not obey the relationship gives an indication at which point a relationship ceases to be truegives an indication at which point a relationship ceases to be true used to determine the constants in an equation relating two variablesused to determine the constants in an equation relating two variables

5. Mr. G DP Physics You need to be able to give a qualitative physical interpretation of a particular graph You need to be able to give a qualitative physical interpretation of a particular graph e.g. as the potential difference increases, the ionization current also increases until it reaches a maximum at….. e.g. as the potential difference increases, the ionization current also increases until it reaches a maximum at….. Graphical Techniques cont…

6. Mr. G DP Physics Plotting Graphs Independent variables are plotted on the x- axis Independent variables are plotted on the x- axis Dependent variables are plotted on the y- axis Dependent variables are plotted on the y- axis Most graphs occur in the 1st quadrant however some may appear in all 4 Most graphs occur in the 1st quadrant however some may appear in all 4

7. Mr. G DP Physics Plotting Graphs - Choice of Axis When you are asked to plot a graph of a against b, the first variable mentioned is plotted on the y axis When you are asked to plot a graph of a against b, the first variable mentioned is plotted on the y axis Graphs should be plotted by hand Graphs should be plotted by hand

8. Mr. G DP Physics Plotting Graphs - Scales Size of graph should be large, to fill as much space as possible Size of graph should be large, to fill as much space as possible choose a convenient scale that is easily subdivided choose a convenient scale that is easily subdivided

9. Mr. G DP Physics Plotting Graphs - Labels Each axis is labeled with the name and symbol, as well as the relevant unit used Each axis is labeled with the name and symbol, as well as the relevant unit used The graph should also be given a descriptive title The graph should also be given a descriptive title

10. Mr. G DP Physics Plotting Graphs - Line of Best Fit When choosing the line or curve it is best to use a transparent ruler When choosing the line or curve it is best to use a 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 every point The line or curve does not have to pass through every point Do not assume that all lines should pass through the origin Do not assume that all lines should pass through the origin Do not do dot to dot! Do not do dot to dot!

11. Mr. G DP Physics y x

12. Analysing the Graph Often a relationship between variables will first produce a parabola, hyperbole or an exponential growth or decay. These can be transformed to a straight line relationshipOften a relationship between variables will first produce a parabola, hyperbole or an exponential growth or decay. These can be transformed to a straight line relationship General equation for a straight line isGeneral equation for a straight line is y = mx + cy = mx + c y is the dependent variable, x is the independent variable, m is the gradient and c is the y- intercepty is the dependent variable, x is the independent variable, m is the gradient and c is the y- intercept The parameters of a function can also be obtained from the slope (m) and the intercept (c) of a straight line graph The parameters of a function can also be obtained from the slope (m) and the intercept (c) of a straight line graph

13. Mr. G DP Physics Gradients (Slope) Gradient = vertical run / horizontal run Gradient = vertical run / horizontal run or gradient =  y /  x or gradient =  y /  x uphill slope is positive and downhill slope is negative uphill slope is positive and downhill slope is negative Don´t forget to give the units of the gradient Don´t forget to give the units of the gradient

14. Mr. G DP Physics Areas 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 is work (N x m = J) e.g. on a force displacement graph the area is work (N x m = J) e.g. on a speed time graph the area is distance (ms -1 x s = m) e.g. on a speed time graph the area is distance (ms -1 x s = m) Again, don´t forget the units of the area Again, don´t forget the units of the area

15. Mr. G DP Physics Standard Graphs - linear graphs b A straight line passing through the origin shows proportionality y x y  x y = k x Where k is the constant of proportionality k = rise/run

16. Mr. G DP Physics Standard Graphs - parabola b A parabola shows that y is directly proportional to x 2 y x2x2 y x i.e. y  x 2 or y = kx 2 where k is the constant of proportionality

17. Mr. G DP Physics Standard Graphs - hyperbola b A hyperbola shows that y is inversely proportional to x y 1/x y x i.e. y  1/xor y = k/x where k is the constant of proportionality

18. Mr. G DP Physics Standard Graphs - hyperbola again An inverse square law graph is also a hyperbola An inverse square law graph is also a hyperbola y 1/x 2 y x i.e. y  1/x 2 or y = k/x 2 where k is the constant of proportionality

19. Mr. G DP Physics Non-Standard Graphs You need to make a connection between graphs and equations You need to make a connection between graphs and equations y x If this is a graph of r against t 2 plotted from data having an expected relationship r = at 2 /2 +r 0 where a is a constant Then the gradient is a/2 and the y-intercept is r 0 - it is not the case that r  t 2, it is a linear relationship The intercept is therefore important too