The slope of the tangent to a v-t graph at one point in time will give instantaneous acceleration in the case where there is non uniform (changing) acceleration. Using a method similar to deriving velocity from distance-time graphs, we can obtain values of acceleration from velocity-time graphs. V instВ = v or, using the slope equation we calulte the slope of the tangent (the red line): To obtain the instantaneous velocity, that is, the velocity at one instant (one point in time - say at exactly t 1), one must takeВ the slope of the tangent line that just touchesВ the curve at that point. This gives us the average velocity between the time interval from t 1В -to- t 2

Slope (of secant line) =В Delta d / Delta t The slope of the secant line (the line that cuts the curve at the two intersecting points (d 1,t 1 and d 2, t 2) can be calculated by the usual slope method: To obtain the average velocity between two points in time (say t 1 and t 2) we can draw the secant line between these two points and calculate the slope of the secant line. It is sufficient to say, however that we can still obtain some useful information by relying on graphical analysis techniques. To analyze this function properly one would need to take the first derivative of the function using Calculus. We know that the velocity is changing as time goes on because the slope of this line is not constant and the function (of the formВ y = ax 2В +В bxВ + c) is an increasing function. This graph illustrates the relationship between the position and the time for an object whose velocity is changing with time. Uniformly Accelerated Motion - Variable Speed We note that 'm', the slope of the line (of the form,В y = mx + b)В is constant and can be calculated by several methods.Д«. The slope m of the position-time graph gives the velocity of the objectВ even when the relation between the position (d) and the time (t)В is not a straight line. This is a typical graph of the relationship between positionВ and time for an object moving at constant speed. Uniform Motion - Constant Speed - no acceleration Deviation of trend lines should be discussed in the analysis of results.ДЇurther information relating to uncertainty is provided in Experimental uncertaintyвЂ‹ and errorsвЂ‹.A. A trend line (if used) should lie within the region of the error bars. Students should normally include vertical error bars at times, horizontal error bars may also be appropriate. Representation of uncertainties in graphsДЄbsolute uncertainties can be represented on graphs using error bars. Conclusions drawn from data must be limited by, and not go beyond, the data available. Gradients and y-intercepts should be considered in terms of what these may indicate about the relationship between independent and dependent variables. In drawing conclusions they should examine patterns, trends and relationships between variables with the limitations of the data in mind. Students should understand why it is important not to вЂforce data through zeroвЂ™. common types of relationships in physics include linear, power and sometimes exponential.not all experiments will show a correlation between variables.the existence of a correlation does not necessarily establish that there is a causal relationship between two variables.the independent variable is represented on the horizontal axis while the dependent variable is represented on the vertical axis.When drawing graphs, students should note that: sketch graphs (not necessarily on a grid no plotted points labelled axes but not necessarily scaled) can be used to show the general shape of the relationship between two variables.

scattergrams can be used to show an association between two variables.

lines of best fit can be used to illustrate the underlying relationship between variables.line graphs can be used to display data in which both the independent and dependent variables are continuous.pie graphs and bar charts can be used to display data in which one of the variables is categorical.The type of graphical representation used by students will depend upon the nature of the investigation and the type of variables investigated: Graphical representations can be used to more clearly show whether any trends, patterns or relationships exist. Although tables are an effective means of recording data, they may not be the best way to show trends, patterns or relationships. To explain the relationship between two or more variables investigated in an experiment, data should be presented in such a way as to make any patterns and trends more evident.

Physics graphical analysis