Physics
Cambridge IGCSE
Cambridge IGCSE
TOPIC 1A: FORCES and MOTION
1.2b Motion graphs
In this section, you will learn how graphs are used to describe the movement of any object. For this course there are two main types of graph used:
The time measurement on all these graphs is always given on the bottom axis - the x-axis.
As the name implies, this is simply a graph of the distance covered by any moving object, from a starting point at a time of zero. Let's have a look at a few basic rules, by looking at a few examples:
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| This graph shows a body that is stationary. The distance is not changing. It is standing still at the start point. | This graph also shows a body that is stationary. However it is standing still some distance from the start point. |
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| This graph shows a body moving away from the start at a constant velocity. The steeper the line, the faster the velocity. (The dotted line shows a body moving slower than the solid line). | This curved line shows a body accelerating. The line is clearly getting steeper (increasing gradient) so the velocity is increasing. |
Figure 1. Basic Distance-Time Graph Features
Using these graphs, we can determine the movement of a body, and also by taking readings from the graph we can use the maths from the last section to find the velocity. Remember that velocity basically means the same as speed in these graphs. have a go at these questions to check you understand this section:
Questions:
1. A man is taking his dog for a walk. When he opens the front door, the dog runs off in a straight line at a steady speed. It then stops at a lamp post for some time.
Sketch a distance-time graph for this motion. You do not need to estimate any values for this question.
Here is an example:
2. A student writes a program to make a small robot move across the desk. This graph shows the motion of the robot:
a) Line 'X' shows the robot moving away from the start at constant velocity.
b) Line 'Y' shows the robot stops / is stationary (100 cm from the start).
c) Line 'Z' shows the robot moving at constant velocity just as line 'X', but this time at a slower velocity as the line is less steep. (Lower gradient).
d) The fastest speed is where the line is the steepest, (the highest gradient), which is line 'X'. For this section, reading from the graph, the robot covers 100 cm in 5 seconds. As we are using cm, the speed will be given in cm/s.
| speed = | distance | = | 100 cm |
| time | 5 s |
so the speed, or velocity, is = 20 cm /s.
( We could convert the distance measurement to metres to give a velocity of 0.2 m/s)
These graphs are more frequently used in exams, as they are slightly harder and give more information about the motion of the body. They are often called speed- time graphs. Again, it is simply a plot of the velocity/speed on the y-axis against time on the x-axis.
Here are the 4 most important features you need to learn on these graphs:
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| This graph shows a body that is stationary. The velocity is zero so it is not moving. | This graph shows a body that is moving with a constant velocity. |
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| This graph shows a body that is speeding up, or accelerating. The steeper the line, the higher the acceleration. (The solid line shows a higher acceleration than the the dotted line). | This graph shows a body that is slowing down, or decelerating. (You can also describe this a negative acceleration). |
Figure 2. Basic Velocity-Time Graph Features
As you can see, these graphs look very similar to the distance-time graphs and they are easily confused. Take care to check the label on the y-axis to make sure you know which is which. Some people just seem to have a knack for understanding what any motion graph is showing. If you find this more difficult, you need to learn these shapes.
As shown above, deceleration can be described as a negative acceleration. So if a car is braking, we can say that is decelerating at (for example) 2 m/s2, or say that it is accelerating at -2 m/s2.
A velocity-time graph is often used to show motion because you can use it to find other information. The two that we will cover here are how to find the acceleration and the distance travelled from the graph.
The distance travelled = the area under the line.
The acceleration = the gradient of the line.
Take this example, showing a cyclist accelerating way from traffic lights:
Figure 3. An Accelerating Cyclist
The line in this graph slopes upwards, showing acceleration. We can use the information to calculate the distance travelled in this time and also the acceleration:
Distance: This is the area under the line (between the red line and the x-axis), shown by the shaded area in this example. The area of this triangular shape is:
Area = ½ x base x height = ½ x 30 x 6
So the distance travelled = 90 m.
(The units should be 'm' because the velocity is given in 'm/s').
Acceleration: This is the gradient of the line. figure 2 above shows that the steeper the line, the higher the acceleration. The 'steepness' of a line is the gradient. From your maths lessons, you should know that:
| gradient = | 'rise' | or gradient = | change in y-axis |
| 'run' | change in x-axis |
On the example above shown in figure 3, the change in the y-axis is 6 m/s, and the change in the x-axis is 30 s. Therefore:
| gradient = | 6 | = 0.2 |
| 30 |
So the acceleration = 0.2 m/s2.
Extension: If you want to find out why these calculations work, click the button here to find our more. You don't need to learn this, but if you find it interesting, read on.....
This section covers the maths behind the meaning of the gradient and area under the line on a velocity-time graph.
1. Acceleration:
The formula for acceleration is:
| acceleration = | change in velocity |
| time taken |
| acceleration = | change in velocity | = | change in y-axis | = gradient |
| time taken | change in x-axis |
2. Distance:
Have a look at this graph:
We know that distance travelled = speed x time.
However in this graph:
speed = velocity = height of the grey shaded box;
time = length of grey shaded box.
Therefore (distance = speed x time) becomes (distance = height x width) which is the area of the box!
So distance travelled = area under the line.
This is true for all shapes - just find the area under the line showing the velocity of the object.
Now have a go at some questions about velocity-time graphs:
Questions:
The following graph is for use with questions 3 & 4, and shows the motion of a tram:
Figure 4. A Velocity time graph for a tram.
3. The graph shown in figure 4 gives data on the movement of a city tram moving away from a station.
Which section or sections of the graph shows the tram ...
4. Using the graph shown in figure 4 for the velocity of a tram, calculate:
a) The distance travelled between 30 and 40 s is given by:
Distance = area under line = rectangle of height 15 (m/s) and length 10 (s).
Therefore the area = distance = 15 x 10 = 150 m.
b) The distance travelled in the first 10 s is given by:
Distance = area under line = area of triangle = ½ base x height,
So area =
½ x 10 x 5,
Area = distance = 25 m.
| acceleration = | change in velocity |
| time taken |
| or by using acceleration = gradient = | change in y-axis |
| change in x-axis |
Both methods give:
| acceleration = | 15 - 5 | = | 10 |
| 10 | 10 |
So the acceleration of section R = 1 m/s2.
A cyclist takes part in a sprint event on a track. At the start of the race, the cyclist produces as large a force as possible to make the bike accelerate. However as he goes faster and faster, air resistance and friction push backwards on the cyclist and bike.
Have a look at the diagram below to see exactly what is happening. In this example, we have ignored friction as a negligible force, and only included the force from the cyclist and air resistance:
Figure 5. A cyclist reaching terminal velocity
Let's look at each stage of figure 5 in a bit more detail:
All of this can be summarised in a velocity - time graph:
Figure 6. A terminal velocity graph
This is a very famous graph and is often used in the exams - you will need to learn how to draw it and explain each part of the process of reaching terminal velocity. Other common examples include skydivers, and other falling objects like squash balls - all reach a different terminal velocity depending on how air resistance affects them. For a falling object, the downward force is the weight (the force of gravity), and air resistance acts upwards.
Remember that liquids also add a resistive force, so this graph could be used for a ball falling through water or any number of experiments where objects move through a gas or a liquid.
Questions:
5. The diagrams below (figure 4) show forces acting on a skydiver who has jumped out of a plane. There are 3 diagrams, showing different stages of the descent.
Figure 7. The forces acting on a skydiver
When any object falls under gravity on Earth, it always initially accelerates downwards at 9.8 m/s2. This is called the acceleration of gravity on Earth (given the letter 'g'), and is independent of the mass of the object.
You can drop a heavy ball or a feather, and IF there is no air resistance, they will both accelerate downwards at g= 9.8 m/s2 as shown in this video. Note that the final fall in the video is in slow motion!
YouTube video - The world's largest vacuum chamber (Human Universe - BBC)
In summary, learn that the acceleration of gravity on Earth (g) = 9.8 m/s2. Note that this value is also called the 'gravitational field strength', as explained in the next section 1.3 on Mass and Weight.
Now test your understanding using these quick, 10 minute questions on this topic from Grade Gorilla:
