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1.3Graphs of motion
If the acceleration is constant the equations derived in SectionÂ 1.2 hold and we can use them to analyse a motion problem. But if acceleration is not constant
these equations do not apply and cannot be used. But we can still analyse a motion using graphs. Our main tools will be those provided in Table 1.1.
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Consider the graph of position against time shown in Figure 1.19.

Figure 1.19: Position against time for accelerated motion.

We would like to predict the shape of the graph of velocity against time. We use the fact that velocity is the gradient of the position-against-time graph. We see that at t = 0 the gradient is positive. As time increases the gradient is still positive but gets smaller and smaller and at t = 0.25 s it becomes zero. It then becomes negative. It gets more and more negative, reaching its most negative value at t = 0.5 s. It remains negative past 0.5 s but the curve gets less steep and the gradient becomes zero again at 0.75 s. From then it becomes positive and increases until 1.0 s. These observations lead us to the velocity-against-time graph shown in Figure 1.20. Note that it is the shape we are after, not detailed numerical values of velocity; hence there is no need to put numbers on the vertical axis.

Figure 1.20: The velocity graph corresponding to the graph of Figure 1.19.

Working in exactly the same way we can find the shape of the acceleration-against-time graph by examining the gradient of the velocity graph we just obtained. The result is Figure 1.21.

Figure 1.21: Acceleration graph corresponding to the velocity graph of Figure 1.20.

CHECK YOURSELF 4
Look at the motion in Figure 1.22.

Figure 1.22

aState the time(s) at which the acceleration is zero.
bState the time(s) at which the acceleration is negative.
cAt what time is the acceleration most negative?
dAt 2.3 s, do you expect the displacement to be positive or negative and why?
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TEST YOUR UNDERSTANDING
19The graph in Figure 1.23 shows the variation of the position of a moving object with time. Draw the graph showing the variation of the velocity of the object with time.

Figure 1.23

20The graph in Figure 1.24 shows the variation of the position of a moving object with time. Draw the graph showing the variation of the velocity of the object with time.

Figure 1.24

21The graph in Figure 1.25 shows the variation of the position of a moving object with time. Draw the graph showing the variation of the velocity of the object with time.

Figure 1.25

22The graph in Figure 1.26 shows the variation of the velocity of a moving object with time. Draw the graph showing the variation of the position of the object with time. The initial position is zero.

Figure 1.26

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23The graph in Figure 1.27 shows the variation of the velocity of a moving object with time. Draw the graph showing the variation of the position of the object with time (assuming the initial position to be zero).

Figure 1.27

24The graph in Figure 1.28 shows the variation of the velocity of a moving object with time. Draw the graph showing the variation of the acceleration of the object with time.

Figure 1.28

25The graph in Figure 1.29 shows how acceleration varies with time.

Figure 1.29

Draw a graph to show how velocity varies with time. The initial velocity is zero.
26The graph in Figure 1.30 shows how acceleration varies with time.

Figure 1.30

Draw a graph to show how velocity varies with time. The initial velocity is zero.
27The graph in Figure 1.31 shows the position against time of an object moving in a straight line. Four points on this graph have been selected.

Figure 1.31

aIs the velocity between A and B positive, zero or negative?
bWhat can you say about the velocity between B and C?
cIs the acceleration between A and B positive, zero or negative?
dIs the acceleration between C and D positive, zero or negative?
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28Sketch velocityâtime plots (no numbers are necessary on the axes) for the following motions.
aA ball is dropped from a certain height and bounces off a hard floor. The speed just before each impact with the floor is the same as the speed just after impact. Assume that the time of contact with the floor is negligibly small.
bA cart slides with negligible friction along a horizontal air track. When the cart hits the ends of the air track it reverses direction with the same speed it had right before impact. Assume the time of contact of the cart and the ends of the air track are negligibly small.
cA person jumps from a hovering helicopter. After a few seconds she opens a parachute. Eventually she will reach a terminal speed and will then land.