4.5 The signed area under a basic velocity-time graph

We have already seen (in ar 3.6) the in the context of uniform motion, the signed area under a particle"s velocity-time graph, between two provided times, to represent the adjust in the particle"s position throughout that time interval, through a hopeful area equivalent to displacement in the confident direction. In the case of uniform motion, the velocity-time graph was a horizontal line and also the area under the graph to be rectangular.

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Now, in the context of non-uniform motion, the seems herbal to questioning if the very same interpretation deserve to be given to the area under a basic velocity-time graph, such together that in between t1 and t2 in figure 33. The answer come this is a definite yes, despite a rigorous evidence is past the border of this course, so what complies with is merely a plausibility argument.

In figure 33, the colour-shaded area under the graph does no take the form of a rectangle. However, it might be approximately represented through a amount of rectangle-shaped areas, as shown in number 34. To create Figure 34 we have broken the time between t1 and also t2 right into a variety of small intervals, each of the same duration Δt and within each little interval, the velocity has actually been approximated through a constant, which deserve to be taken to be the median velocity throughout that interval. Together a result, the area of each rectangle-shaped strip in figure 34 represents the approximate change of position over a brief time Δt and also the amount of those areas represents the approximate adjust of position over the interval t1 to t2. Now, if us were to repeat this process while making use of a smaller value for Δt, as in number 35, then we would certainly have an ext strips between t1 and also t2; their full area, representing the approximate readjust in position in between t1 and t2, would be an also closer approximation to the true area shown in figure 33. Offered these results, it appears reasonable to mean that if we enabled Δt to end up being smaller and also smaller, if the number of rectangular strips between t1 and t2 came to be correspondingly larger and also larger, then we would certainly eventually uncover that the area under the graph in figure 33 was precisely equal come the adjust in position between t1 and t2.


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Figure 34: The area under the velocity-time graph of figure 33, damaged up into thin rectangle-shaped strips
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Figure 35: The area under the velocity-time graph of figure 33, damaged up into even much more rectangular strips by reduce the value of Δt

This conclusion is in fact correct and can be confirmed in a rigorous means by considering what mathematicians speak to a limit, in this instance "the limit together Δt often tends to zero". We shall not pursue that here, however we should note that it wasn"t until the beforehand nineteenth century that the mathematics of limits was properly formulated, back it was in use long before then. That is likewise worth stating that it was the advance of the idea the a border that ultimately laid Zeno"s paradox come rest. Just as the increasing variety of diminishing strips have the right to have a finite full area, for this reason the increasing number of smaller steps that Achilles need to take to with the tortoise deserve to have a limited sum and be completed in a limited time. Rigorous mathematical reasoning agrees with our everyday experience in informing us that motion have the right to exist and that athletes outrun tortoises!


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Figure 36 reflects the velocity-time graph for a bit with constant acceleration.

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(a) What is the displacement the the fragment from that is initial position after 6 s?

(b) What is the distance took trip by the particle in between t = 2 s and t = 6 s? (You may find it helpful to recognize that the formula because that the area that a triangle is: area = half the base × height, and also that because that the area the a trapezium is: area = base × (half the amount of the lengths the the parallel sides).)


Answer

(a) The displacement over the an initial 6 s is equal to the full signed area in between the graph and the t-axis and between t = 0 s and also t = 6 s in figure 37. Recall the regions listed below the axis are concerned as an adverse areas. In this instance the full area is composed of two triangles, one over the axis, one below. The full displacement is thus given by

sx(6 s) = (1/2)×(1 s)×(1.2 m s−1) −(1/2)×(5 s)×(6 m s−1)

i.e. Sx(6 s) = (0.60 m) −(15 m) = −14.4 m.

(b) The street travelled between t = 2 s and also t = 6 s will be the size of the displacement over that time. Note that the displacement will be an adverse since that is stood for by the area of the colour-shaded trapezium, i m sorry is entirely below the axis. However, the corresponding distance will certainly be optimistic (since it is a magnitude) and will have actually the value