Introduction


Information about movement can be presented in a number of ways. Data for the total distance moved at different times during movement can be recorded in a table. Alternatively, the same information can be presented in a graph.





The table and graph both present information about an object's position at certain times. In this unit we will be looking at how such data can be interpreted and how additional quantities can be determined from graphs.

A number of different techniques can be used for recording the way distance changes over time. One method commonly used in the school laboratory is the ultrasonic motion

sensor

Sensors used in electronics produce a change in their resistance when some feature of their surrounding environment changes. The resistance of a thermistor changes as the surrounding temperature alters.

sensor.



This device emits a pulse of ultrasound which travels through the air at 330 ms⁻¹ before rebounding off a target. The distance of the reflecting object from the sensor can be found by measuring the time interval between the emitted and reflected pulses and using the following equation:



The distance can be measured at regular time intervals, as the trolley moves, and a graph plotted. This type of measurement is best done by a computer where the rapid timings and calculations can be done quickly and results displayed on the monitor.

Other techniques are available for detecting distance and speed. A 'radar gun' emits a pulse of

electromagnetic

Electromagnetic waves, such as light, are made up from oscillating electric and magnetic fields. Because of this, they are self-propagating and can travel through a vacuum. All types of electromagnetic wave travel at the same speed in a vacuum, 3 × 10⁸ ms⁻¹.

electromagnetic radiation with a certain

frequency

The wave frequency f is the number of complete waves passing any point each second. Frequency is measured in hertz, Hz.

frequency. A passing vehicle will reflect the pulse back to a detector on the gun. The frequency of the reflected radiation will be dependent upon the speed of the moving reflector.

The speed of the moving reflector can be determined by measuring differences between the emitted and reflected pulses.

The graph in Fig.2 shows a set of distance and time axes. Its y-axis shows the horizontal distance of the helicopter from its starting position. (This is indicated on the animation by the length of the line between the take-off and landing positions).

Start the helicopter in Fig.2 flying and drag it horizontally towards the landing pad.


As the helicopter flies towards the landing pad, the distance from its starting point increases and so the line on the graph moves from the bottom left-hand side of the axes towards the top right.

Use the animation in Fig.2 to answer the following questions.



A line drawn through the points on a graph, such as that shown in Fig.3, allows us to see trends. We can also use the line in the graph to estimate values that would lie between data points given in a results table, so a graph of Fig.3 contains more information than the table below.





Clearly in Fig.3 the helicopter is moving away from the starting point so the graph has a positive gradient. Additionally, the distance from the reference point increases in equal steps as time passes. This means that the graph is a straight line and therefore has a constant or uniform gradient.

The size of the gradient shows how quickly the distance travelled is changing with time. In this example the distance is changing at a rate of 10 m every second. Consequently we can say that this graph records the motion of an object moving with a constant speed of 10 ms⁻¹.

The balls shown in Fig.5 are moving with different speeds but the speed of each is constant.


Click on the figure below to interact with the model.




A straight line on a distance–time graph indicates that an object is moving at a steady speed. The steeper the line, the faster the speed. Both balls in the simulation of Fig.5 are moving with a steady speed and in a fixed direction. Therefore the velocities as well as the speeds are constant. However, in order to state the

velocity

An object's velocity states both the speed and direction of motion relative to a fixed reference point.

velocity we would have to indicate the direction of movement in addition to the speed.

Grab the ball in the simulation of Fig.6 and drag it to the top of its range of motion. Release the ball and observe its motion and the distance versus time graph as it falls.


Click on the figure below to interact with the model.




The distance values in the graph of Fig.6 decrease as time passes because the ground is being used as the point from which distances are measured. The ball falling in Fig.6 is moving towards the reference point, so the measured distances are decreasing with time.

Fig.7 shows the distance–time graph for an object speeding up as it moves away from its reference point. The curve of the line shows that the ball is getting faster as time passes. To find the speed at any instant during this motion we would have to draw a tangent to the curve and determine its gradient.



On distance–time graphs, bigger speeds are indicated by steeper lines. We can regard the curve in Fig.7 as being made up from a large number of short straight lines each having a slightly different gradient.

The curve in Fig.7 gets steeper as the distance increases, showing that the object is moving progressively faster as it moves away from the observation point. This object is accelerating away from its starting position.

When the mass in Fig.8 falls the trolley accelerates and the distance–time graph for the

acceleration

An object's acceleration is its rate of change of velocity.

acceleration is plotted.



By clicking on the right-hand green button in Fig.8, you can switch from a trolley towed by a falling mass to one towed by a falling chain. The falling mass and the falling chain produce different distance–time graphs because they represent different types of acceleration.

When the trolley is being towed by the falling mass its speed increases uniformly. The falling mass produces a constant acceleration. As the chain falls, the towing force increases and the increase in the speed each second is not uniform. The falling chain produces an increasing acceleration.

In most of the situations we will meet, movement is in a straight line and so displacements from the starting point are easy to calculate. The label on the y-axis of Fig.9 shows that an increase in the y-axis value represents motion due north.



The object whose motion is represented by the graph in Fig.10 moves north at a steady speed of 3 ms⁻¹ for 2 seconds. After this time the

displacement

An object's displacement quotes both its bearing and distances relative to a fixed reference point.

displacement decreases as the object starts to move in the opposite direction. After 6 seconds the object is just 3 m north of its starting point even though it has travelled a total distance of 9 m.



The object's velocity after 6 seconds is determined from the gradient of the line at that point. The size of the gradient is 0.75 ms⁻¹, so after 6 seconds the velocity of the object is 0.75 ms⁻¹ in a southerly direction.

In the set-up of Fig.11 the position of the trolley relative to the motion sensor is plotted. Move the trolley to match the motion described on the preset graph drawn and then answer the questions below. There are a number of preset graphs that you might want to investigate.

Summary


Drawing a line on a graph shows the trends in data presented in a table. Additional information can be determined from the slope of the graph.

The size of the slope or the gradient of a distance versus time graph gives the speed at which an object is moving.

The gradient of a displacement versus time graph indicates an object's velocity.