Distance=E2=80=93Time Graphs

Introduction

Information about moveme= nt can=20 be presented in a number of ways. Data for the total distance moved at dif= ferent=20 times during movement can be recorded in a table. Alternatively, the same= =20 information can be presented in a graph.

Time / s	Distance from = start /=20 m
0	0
2	20
4	40
6	60
8	80

The=20 table and graph both present information about an object's position at cer= tain=20 times. In this unit we will be looking at how such data can be interpreted= and=20 how additional quantities can be determined from graphs.

Motion sensors

A number of different techniqu= es can=20 be used for recording the way distance changes over time. One method commo= nly=20 used in the school laboratory is the ultrasonic motion

sensor

= Sensors used=20 in electronics produce a change in their resistance when some feature of t= heir=20 surrounding environment changes. The resistance of a thermistor changes as= the=20 surrounding temperature alters.

sensor.

Figure 1. Using= a=20 motion sensor.

This de= vice emits a=20 pulse of ultrasound which travels through the air at 330 ms^=E2=
=88=921=20 before rebounding off a target. The distance of the reflecting object from= the=20 sensor can be found by measuring the time interval between the emitted and= =20 reflected pulses and using the following equation:

1. How long will a pulse of ultraso= und=20 take to travel from the emitter to the trolley if the trolley= is=20 1.65 m away from the motion sensor?

	2.5 ms
	5 ms
	10 ms
	20 ms

2. What is the approximate delay be= tween=20 emitted and reflected pulses when the trolley is 1.65 m away from th= e=20 emitter?

	2.5 ms
	5 ms
	10 ms
	20 ms

The distance=20 can be measured at regular time intervals, as the trolley moves, and a gra= ph=20 plotted. This type of measurement is best done by a computer where the rap= id=20 timings and calculations can be done quickly and results displayed on the= =20 monitor.

Other techniques are available for detecting distance and = speed.=20 A 'radar gun' emits a pulse of electromagnetic radiation with a certain=20 frequency. A passing vehicle will reflect the pulse back to a detector on = the=20 gun. The frequency of the reflected radiation will be dependent upon the s= peed=20 of the moving reflector.

The speed of the moving reflector can be= =20 determined by measuring differences between the emitted and reflected=20 pulses.

Distance=E2=80=93time graphs

The graph in Fig= .2 shows a set=20 of distance and time axes. Its y-axis shows the horizontal distance of the= =20 helicopter from its starting position. (This is indicated on the animation= by=20 the length of the line between the take-off and landing positions).=20

Start the helicopter in Fig.2 flying and drag it horizontally towa= rds=20 the landing pad.

Figure 2. Helic= opter=20 motion

As the helicopt= er flies=20 towards the landing pad, the distance from its starting point increases an= d so=20 the line on the graph moves from the bottom left-hand side of the axes tow= ards=20 the top right.

Use the animation in Fig.2 to answer the following=20 questions.

3. The horizontal distance between = the=20 take-off and landing pads is approximately =E2=80=A6

	10 m.
	14 m.

4. If the helicopter moves quickly = from=20 the take-off position to the landing pad the line on the graph has=20 =E2=80=A6

	a steep gradient.
	a negative gradient.

5. If during the journey the helico= pter=20 returns partly back towards the starting point the graph for that pa= rt of=20 the journey has =E2=80=A6

	a positive gradient.
	a negative gradient.

Figure 3. Graph= for a=20 helicopter moving at constant=20 speed.

A line drawn through = the=20 points on a graph, such as that shown in Fig.3, allows us to see trends. W= e can=20 also use the line in the graph to estimate values that would lie between d= ata=20 points given in a results table, so a graph of Fig.3 contains more informa= tion=20 than the table below.

<= /TR>

Time / s	Distance from start / m
0	0
2	20
4	40
6	60
8	80

<= /TD>

Figure 4. Distance vers= us time=20 data.

Clearly in F= ig.3 the=20 helicopter is moving away from the starting point so the graph has a posit= ive=20 gradient. Additionally, the distance from the reference point increases in= =20 equal steps as time passes. This means that the graph is a straight= line=20 and therefore has a constant or uniform gradient.

The size of the=20 gradient shows how quickly the distance travelled is changing with time. I= n this=20 example the distance is changing at a rate of 10 m every second.=20 Consequently we can say that this graph records the motion of an object mo= ving=20 with a constant speed of 10 ms^=E2=88=921.

The balls= shown in=20 Fig.5 are moving with different speeds but the speed of each is=20 constant.

Click on the figure below to interact with the=20 model.

Figure 5. Bounci= ng=20 balls.

6. Which ball in Fig.5 is moving=20 faster?

	Red
	Blue

7. Complete the following statement= s to=20 summarize the behaviour of the balls in Fig.5.

The fact that the graph for each ball is a

line=20 indicates that the speed of each ball is

. The f= aster=20 ball can be identified by the line with the

=20 gradient.

A straight=20 line on a distance=E2=80=93time graph indicates that an object is moving a= t a steady=20 speed. The steeper the line, the faster the speed. Both balls in the simul= ation=20 of Fig.5 are moving with a steady speed and in a fixed direction. Therefor= e the=20 velocities as well as the speeds are constant. However, in order to state = the=20

velocityAn=20 object's velocity states both the speed and direction of motion relative t= o a=20 fixed reference point.

velocity we would have to indicate the= =20 direction of movement in addition to the speed.

Grab the ball in th= e=20 simulation of Fig.6 and drag it to the top of its range of motion. Release= the=20 ball and observe its motion and the distance versus time graph as it falls= .=20

Click on the figure below to interact with the model.

Figure 6. Droppi= ng=20 ball.

8. When the ball is falling towards= the=20 ground, the distance values plotted on the graph are =E2=80=A6

	increasing.
	decreasing.

The distance=20 values in the graph of Fig.6 decrease as time passes because the ground is= being=20 used as the point from which distances are measured. The ball falling in F= ig.6=20 is moving towards the reference point, so the measured distances are decre= asing=20 with time.

Finding the speed

Fig.7 shows the distance=E2= =80=93time graph=20 for an object speeding up as it moves away from its reference point. The c= urve=20 of the line shows that the ball is getting faster as time passes. To find = the=20 speed at any instant during this motion we would have to draw a tangent to= the=20 curve and determine its gradient.

Figure 7. Findi= ng the=20 speed at any instant during the=20 motion.

9. In Fig.7 the speed of the object= 6=20 seconds after the start of the motion is approximately=20 =E2=80=A6

	0.59 ms^=E2=88=921.
	0.94 ms^=E2=88=921.

10. When the object in Fig.7 is 35 = m away=20 from its starting position, it is travelling with a speed of approxi= mately=20 =E2=80=A6

	1.48 ms^=E2=88=921.
	0.52 ms^=E2=88=921.

11. The object in Fig.7 is moving a= t a=20 speed of approximately 1 ms^=E2=88=921 after travelli= ng for a time=20 of =E2=80=A6

	6.4 s.
	2.4 s.

On=20 distance=E2=80=93time graphs, bigger speeds are indicated by steeper lines= . We can=20 regard the curve in Fig.7 as being made up from a large number of short st= raight=20 lines each having a slightly different gradient.

The curve in Fig.= 7 gets=20 steeper as the distance increases, showing that the object is moving=20 progressively faster as it moves away from the observation point. This obj= ect is=20 accelerating away from its starting position.

When the mass in Fig= .8=20 falls the trolley accelerates and the distance=E2=80=93time graph for the =

acceleration=

An=20 object's acceleration is its rate of change of velocity.=20

acceleration is plotted.

Figure 8. An=20 increasing acceleration.

<= BR>By=20 clicking on the right-hand green button in Fig.8, you can switch from a tr= olley=20 towed by a falling mass to one towed by a falling chain. The falling mass = and=20 the falling chain produce different distance=E2=80=93time graphs because t= hey represent=20 different types of acceleration.

When the trolley is being towed by= the=20 falling mass its speed increases uniformly. The falling mass produces a co= nstant=20 acceleration. As the chain falls, the towing force increases and the incre= ase in=20 the speed each second is not uniform. The falling chain produces an increa= sing=20 acceleration.

Displacement=E2=80=93time graphs

In most of t= he situations we=20 will meet, movement is in a straight line and so displacements from the st= arting=20 point are easy to calculate. The label on the y-axis of Fig.9 shows that a= n=20 increase in the y-axis value represents motion due north.

Figure 9. Displacement= =E2=80=93time=20 graph.

12. How far north does the object m= ove in=20 the first 2 seconds?

	6 m
	2 m

13. Calculate the average speed of = the=20 moving object during the first 3 seconds.

<= /TABLE>

	0.33 ms^=E2=88=921
	3 ms^=E2=88=921

14. Three seconds after the beginni= ng of=20 the motion of Fig.9 the object is =E2=80=A6

	travelling due north.
	travelling due south.
	travelling south-east.

15. How far is the object away from= its=20 starting position six seconds after the start of its=20 motion?

	9 m
	6 m
	3 m

Figure 10. Displacement= =E2=80=93time=20 graph.

The object whos= e motion is=20 represented by the graph in Fig.10 moves north at a steady speed of=20 3 ms^=E2=88=921 for 2 seconds. After this time the

displacement=

An=20 object's displacement quotes both its bearing and distances relative to a = fixed=20 reference point.

displacement decreases as the object starts = to=20 move in the opposite direction. After 6 seconds the object is just 3 = m=20 north of its starting point even though it has travelled a total distance = of=20 9 m.

16. The average speed of the object= whose=20 motion is represented by the graph of Fig.10 for its entire journey = is=20 =E2=80=A6

	0.75 ms^=E2=88=921.
	1.5 ms^=E2=88=921.
	6 ms^=E2=88=921.

The object's=20 velocity after 6 seconds is determined from the gradient of the lin= e at=20 that point. The size of the gradient is 0.75 ms^=E2=88=921,= so after 6=20 seconds the velocity of the object is 0.75 ms^=E2=88=921 in= a southerly=20 direction.

Match graphs

In the set-up of Fig.11 the posi= tion of=20 the trolley relative to the motion sensor is plotted. Move the trolley to = match=20 the motion described on the preset graph drawn and then answer the questio= ns=20 below. There are a number of preset graphs that you might want to=20 investigate.

Figure 11. Matc= h=20 graphs.

17. Complete the following=20 statements.

Moving the trolley to the left causes the measured dista= nce=20 to

while motion to the right causes the distanc= e to=20

. Slow motion produces a line with a

slope=20 while fast movement produces a

line. Motion to = the left=20 produces a line with a

gradient and motion to t= he right=20 results in a line with a

=20 gradient.

Summary

Drawing a line on a graph sho= ws the=20 trends in data presented in a table. Additional information can be determi= ned=20 from the slope of the graph.

The size of the slope or the gradient = of a=20 distance versus time graph gives the speed at which an object is=20 moving.

The gradient of a displacement versus time graph indicates = an=20 object's velocity.

Exercises

Figure 12.

18. Fig.12 shows the speed of a tra= in as=20 it travels along a certain section of track.

What time does the train take to travel = along=20 this section of track?

s &nb= sp;(to=20 the nearest whole number)

What is the total length = of=20 this section of track?

m &nb= sp;(to=20 the nearest whole number)

What is the average speed= of=20 the train during its journey along this section of track?
<= IMG=20 src=3D""> ms^=E2=88=921 (to 1= =20 d.p.)

Figure 13.

19. Fig.13 shows the speed=E2=80=93= time graph for=20 an object starting from rest and increasing its speed as it=20 travels.

What is the speed of the object 5 second= s=20 after starting?
ms^=E2=88=921 (to th= e nearest whole=20 number)

What is the average speed over the first 5= =20 seconds?

ms^=E2=88=921 &= nbsp; (to 1=20 d.p.)

What distance does the object travel during t= he=20 first 5 seconds?
ms^=E2=88=921 (to 1= =20 d.p.)

What is its average speed between times of 5 = and 10=20 seconds?

ms^=E2=88=921 &= nbsp; (to=20 the nearest whole number)

Figure 14.

20. Fig.14 shows a distance=E2=80= =93time graph=20 for a toy car being pushed uphill towards an observer.=

How far is the observer from the point w= here=20 the car starts?

m (to = the=20 nearest whole number)

How far does the car travel b= etween=20 2 seconds and 6 seconds after release?
m (to the nearest whole=20 number)

Figure 15.

21. In an experiment, as shown in F= ig.15,=20 a ball, held directly underneath a motion sensor, is released. A pup= il=20 makes the following statements about the graph. Decide whether they = are=20 true or false.
	At point P the ball is held directly underneath the mo= tion=20 sensor.
	Between Q and R the ball is in contact with the ground= .
	Between S and T the ball is rising.

There are restrictions on the use of this coursew= are.=20
Please click here for terms and conditions.

Well done!

Try again!

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}, setTop: function ( yPos ) { this.theObject.top =3D yPos; }, getTop: function ( ) { return parseInt(this.theObject.y); }, =09 getAbsTop: function ( ) { var yPos =3D this.theObject.offsetTop; parentEl =3D this.theObject.offsetParent; while (parentEl !=3D null)=20 { yPos +=3D parentEl.offsetTop; parentEl =3D parentEl.offsetParent; } return yPos; }, getAbsLeft: function ( ) { var xPos =3D this.theObject.offsetLeft; parentEl =3D this.theObject.offsetParent; while (parentEl !=3D null)=20 { xPos +=3D parentEl.offsetLeft; parentEl =3D parentEl.offsetParent; } return xPos; }, =09 getWidth: function() { return this.theObject.offsetWidth; }, =09 =09 setPosition: function ( xPos, yPos )=09 {=20 this.theObject.style.left =3D xPos+"px";=20 this.theObject.style.top =3D yPos+"px"; }, switchs: function () {=20 if(this.theObject.style.display=3D=3D"none") {=20 this.theObject.style.display =3D "block";=20 } else {=20 this.theObject.style.display =3D "none";=20 }=20 }, display: function ( which )=20 {=20 if( which!=3D"" ) { this.theObject.style.display =3D which;=20 } else { return this.theObject.style.display; } }, show: function () { this.theObject.style.visibility=3D"visible"; }, hide: function () { this.theObject.style.visibility=3D"hidden"; }, =09 isSelected: function() { return this.theObject.checked; }, setCorrect: function() { this.theObject.src =3D imgCorrect; }, setIncorrect: function() { this.theObject.src =3D imgIncorrect; }, cssName: function() { return this.theObject.getAttribute("class"); } }; =09 // Global access object prototype=20 ccCBA.prototype =3D { getElementById: function ( objId ) { return document.getElementById( = objId ); }, =09 =09 referenceLeft: function() { var reference =3D new ccElement( = 'ccCBAreference' ); return reference.getAbsLeft(); }, =09 referenceRight: function() { var reference =3D new = ccElement('ccCBAreference'); return reference.getAbsLeft() + 590; }, =09 getTargetId: function ( event ) { return (event.target.nodeType = =3D=3D 3) ? event.target.parentNode.id : event.target.id; } }; // Preload QImages var imgIncorrect =3D "layout/incorrect.png"; var imgIncorrectPre =3D = new Image(); imgIncorrectPre.src =3D imgIncorrect; var imgCorrect =3D "layout/correct.png"; var imgCorrectPre =3D new = Image(); imgCorrectPre.src =3D imgCorrect; =09 var imgup =3D new Image(); imgup.src =3D "layout/btndwn.png"; var imgdwn =3D new Image(); imgdwn.src =3D "layout/btnup.png"; } else { alert("This courseware is designed for IE5+/Netscape 6+. \nSome = functionality may not be present in other browsers."); } = //-----------------------------------------------------------------------= -- // Constructors // Constructor for element wrapper function ccElement( objOrId )=20 {=20 if( (typeof objOrId)=3D=3D"string" ) { this.theObject =3D this.localGetElementById( objOrId ); } else { this.theObject =3D objOrId; } } =09 // Constructor for object providing global cross browser functions function ccCBA() { } = //-----------------------------------------------------------------------= ------ // Instantiate the cross browser object var CBA=3Dnull; CBA =3D new ccCBA(); =09 = //-----------------------------------------------------------------------= ------ // Q&A // Check multiple choice answers function check( qtype, qid, cid, range ) { var i, ansimg, temp; var index=3D-1; =09 // First hide all the answers if more than one if(range>1) hideAll( qid, range ); // Find the input the user has selected for(var i=3D1; i<=3Drange; i++) { temp =3D CBA.getElementById( qid+"a"+i ); if( temp.checked ) { index=3Di; break; } } =09 // Check the user actually entered an answer if(index=3D=3D-1) { alert("Please select an answer first."); return; } // Set the indicator ansimg =3D new ccElement( qid+"a"+index+"i" ); if( (qid+"a"+index)=3D=3Dcid ) { // Set the correct icon ansimg.setCorrect(); } else { // Set the incorrect icon ansimg.setIncorrect(); } ansimg.show(); } // Check multiple choice answers function checkMR( qtype, qid, cid, range ) { var i, ansimg, temp; var correct =3D new Array(); correct =3D cid.split(";"); correct =3D myunshift( correct, "dummy" ); //Question elements = numbered from one, arrays from 0, makes the same. =09 // First hide all the answers if more than one if(range>1) hideAll( qid, range ); // For each answer, if the user has selected check if it is right for(var i=3D1; i<=3Drange; i++) { // Get the indicator ansimg =3D new ccElement( qid+"a"+i+"i" ); temp =3D CBA.getElementById( qid+"a"+i ); if( temp.checked ) { // The user has selected this answer, so see if it is correct if(correct[i]=3D=3D"1") { // Set the correct icon ansimg.setCorrect(); } else { // Set the incorrect icon ansimg.setIncorrect(); } } else { // The user has selected this answer, so see if it is correct if(correct[i]=3D=3D"0") { // Set the correct icon ansimg.setCorrect(); } else { // Set the incorrect icon ansimg.setIncorrect(); } } ansimg.show();=09 } } =09 // Check filltheblanks answers function checkFill( qtype, qid, cid, range ) { var i, ansimg; var correct =3D new Array(); correct =3D cid.split(";"); correct =3D myunshift( correct, "dummy" ); //Question elements = numbered from one, arrays from 0, makes the same. // First hide all the answers if more than one if(range>1) hideAll( qid, range ); // For each sub answer, check against corrects for(var i=3D1; i