There's also a large error in the difference in x, leading to a very large error in the calculated velocity. The time difference is 0.5В В±В 0.2В s, a maximum error of 40%. For example, consider the first two points on this graph. But what if the velocity varies with time? Can we take the slope of the graph between each two points? The problem with doing so is that a given absolute error in measurements becomes a large proportional error when the difference is small. In the example shown, the data are well fitted by a straight line (given as mВ±dm in the table) whose slope can be determined with reasonable accuracy. This is shown in red on the graph at right. How can we show the point (t,x) = (1.3В В±В 0.1В s, 4.1В В±В 0.2В m) on a graph? We do this by drawing a cross, whose vertical axis goes from 3.9 to 4.3В m and whose horizontal axis goes from 1.2 to 1.4В s. (If we had done a detailed study of our timing reproducibility and found its standard error to be 0.15В s, we could write t = 1.32В В±В 0.15В s.) To make the error explicit, we might write t = 1.3В В±В 0.1В s. Consequently, this should be written (as we say) to two significant figures, t = 1.3В s. So, to avoid misleading the reader, you should retain only the two significant figures. You are implying a precision that you don't actually have. Now, if you write t = 1.32В s, you are implying that the '2' means something, that it is significant. So the measurement error here is (very roughly) about 0.1В s. Repeated measurements of the same period (eg the time it takes a second hand to pass 2 seconds on a clock) will not usually give the same answer, and the variation is probably about 0.1В s. You will probably find, however, that you are not. We might be tempted to write t = 1.32В s, because the watch is capable of measurements with this precision. We should therefore write x = 4.1В В±В 0.2В m. It is difficult to define the position of a person with greater precision than about 0.2В m, because of the relative motion of arms and legs, so the error in the measurement is likely to be roughly this, or greater. As you do so, you click the stopwatch, which shows 1.32 seconds. How accurately are the data known? Suppose you make the measurement x = 4.1В m by looking at a tape measure. Numbers (not quantities) are what we plot on the axes: the axes really are x/m and t/s, so it is a good idea to to label them in this way.Д®rrors, error bars and significant figures This has the advantage that, when x is divided by a metre or t is divided by a second, the result is a number. The method used here is to plot x/m and t/s. The disadvantage with this convention is that it may suggest that x is a function of m, and it is awkard when one really does want to plot x as a function of m. There are two common ways of representing units on the axes of graphs (here m and s). Multiples and submultiples (kilometre, microsecond) are used when needed. In science and engineering, the fundamental units for length and time are metres (abbreviation m) and seconds (s).

Of course physics was still happening before we set our watch, so anything that happens before t = 0 would be represented on the negative part of the time axis. If the watch starts at zero seconds, any time after that is positive time (t > 0).

It could be the time at which we set a stop-watch ticking. The reference for displacement is the wall, x = 0. Formally, the graph shows his position as a function of time. We call this displacement x, where x is positive if he is to the right of the wall. The man's distance from some reference position, here the wall, is how far he is displaced from it. How can we keep track of this fellow? In other words, how do we show his position at any time? The graph above answers this question. The first chapter in Physclips mechanics uses displacement-time and velocity-time graphs for a man walking in a straight line, so we'll begin with this animation. Errors, error bars and significant figures.

Graphs, errors, significant figures, dimensions and units This page supports the Physclips project,

Distance vs time graph physics data