Worksheet 1 D-t Graphs Complete Solutions

Raffles Girls’ School (Secondary) Raffles Programme Year Three Mathematics Name : Class : Sec 3 / Register No : Date : Unit : Kinematics: Distance-Time, Displacement-Time Graphs Type : Worksheet 1

Prepared by CC Koh 1 Adapted from MT Tan

Our Goal To develop good Mathematical Thinkers

who are effective problem solvers and are able to use the language of Mathematics for precise communication.

ESSENTIAL QUESTION What role do distance-time or displacement-time graphs play in real life applications? LESSON OBJECTIVES By the end of this lesson you should be able to: 1 state why graphs are used to depict distance-time or displacement-time relationships 2 analyse distance-time graphs to ascertain suitability in depiction of real life situations 3 construct distance-time graphs given specific data 4 state the significance of gradient of d-t graphs INTRODUCTION What is Kinematics about?

Kinematics is the study of objects in motions, specifically straight-line motion. This motion can either be vertical, as in falling objects or objects projected vertically, or horizontal, such as cars, bikes, trains or people moving in a straight line. In kinematics we look at the relationship between the position, velocity and acceleration of a particle or object. A number of methods can be used to solve problems involving motion in a straight line. These include graphical methods and the use of calculus. Why use graphs?

The world in which we are studying is the physical world - a world which we can see. And if we can see it, we certainly ought to visualize it. If we seek to understand it, then that understanding ought to involve visual representations, such as graphs. ACTIVITY Consider the following situation. Which graphs do you think best represent the events?

A school prefect hoisting the school flag to the top of the flagpole.

heig

ht o

f fla

g

heig

ht o

f fla

g

heig

ht o

f fla

g

heig

ht o

f fla

g

heig

ht o

f fla

g

time time time time time

Acceptable Graphs

Year Three Mathematics: Kinematics WS 1: Distance-Time, Displacement-Time Graphs


DEFINITION Displacement vs Distance

The displacement of a moving object is defined as the change of position relative to a fixed point or the distance from the arbitrary starting point. It is measured in the same units as distance. However, the displacement is not always the same as the distance travelled. Why? This is because displacement is a vector quantity while distance is a scalar quantity. A vector quantity requires both the magnitude and direction while a scalar quantity only has magnitude. Can you give another example to demonstrate the difference in the two? Speed v.s. Velocity Force (Vector) Acceleration (Vector) DISTANCE-TIME VS DISPLACEMENT-TIME GRAPH The shape of a d-t graph

When an object is in motion, its distance travelled and the time taken to cover that distance can be demonstrated in a distance-time graph or displacement-time graph. What would be a suitable name for the vertical axis of each of the following graphs?

The gradient of a d-t graph

How can you interpret the lines or curves in the graphs? The gradient of the lines or curves gives information about the rate of change of distance/displacement. If the graph is a straight line, the rate of change is uniform throughout the period of motion. If the graph is a curve, the rate of change is non-uniform. If negative gradient, then object is slowing down. If positive gradient, then object is speeding up.

Time Time

Time Time

Distance travelled/ Displacement

Distance travelled/ Displacement

Distance from A/ Displacement

Distance from A/ Displacement

A particle starts at O and moves to a point A and then to point B as shown distance travelled = 5+11=16 units displacement from O = -6 units

O AB

5 units-6



2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

Time (minutes)

Distance (km)

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12 Distance from House (km)

Time (minutes)

CLASS WORK

1. A cyclist left his house and cycle to school, which is 12 km away. The table gives the time since leaving his house and the distance travelled. Sketch the graph representing the motion.

Time (min) 0 4 8 12 14 16 20 Distance (km) 0 3 6 10 10 10 12

2. A cyclist left his house and cycle to school, which is 12 km away. The table gives the time since leaving his house and the distance from the house. Sketch the graph representing the motion.

Time (min) 0 4 8 12 14 16 20 Distance (km) 0 3 6 8 11 9 12

Discussion

What are the differences between graphs of Q1 and Q2? Q1 shows total distance traveled while Q2 shows the displacement from the reference point. Although both cyclists are eventually 12 km from the house, the cyclist of Q2 has actually traveled through a larger distance than cyclist of Q1. Determine the units used for the gradient in both Q1 & Q2. Deduce the significance of the

gradients of the two graphs. Units is km/min, which is the same units for speed and velocity. Given that speed is a scalar quantity, it is probably represented by the gradient of Q1. Similarly, gradient of Q2 shows the velocity of the cyclist.



Distance traveled (km)

20

40

60

80

100

120

0820 0840 0900 0920 0940 1000 1020 1040 Time

EXERCISE

1. A motorist left town X at 0800 for town Y, situated 120 km away, travelling at a constant

speed of v km/h, so as to arrive at Y at 1040. But after travelling for 80 minutes, his car developed engine trouble and he had to stop for 30 minutes to repair it. Then he continued his journey at a speed of u km/h, so as to arrive at Y at 1040. (a) Sketch the distance-time graph of the car, (b) Calculate the values of v and u.

a) b) v=45, u=72 2. The graph which consists of straight line segments

represents the distance of a cyclist from his starting point (O) plotted against time.

Find the average speeds in km/h of the cyclist: (a) between 1100 and 1130 (b) on his outward journey (c) on his return journey (d) entire journey At 1130, a pedestrian was 12 km from O and walked towards O at a steady speed of 6 km/h. Find the times of the two occasions when he met the cyclist.

(a) average speed = 8 ÷ 0.5 = 16 km/h

(b) average speed on outwork journey = 12 ÷ 50/60 = 14.4 km/h

(c) average speed on return journey = 12 ÷ 40/60 = 18 km/h

(d) average speed on entire journey = 24 ÷ 100/60 = 14.4 km/h

eq of AB : 251

+= xy ---------(1)

eq of CD : 15101

+−= xy ----(2)

eq of EF : 30103

+−= xy ----(3)

(1)=(2) ∴ hT 11431 =

(2)=(3) ∴ hT 12152 =

time

12

8

11.00 11.30 11.50 12.00 12.40

time

12

8

11.00 11.30 11.50 12.00 12.40

A

B E

F

C

D



3. Two towns P and Q are 120 km apart. At time 0900, a car X leaves P and travels to Q at a steady speed of u km/h where 25 ≤ u ≤ 50. At the same time, at 0900, a car Y leaves Q and travels to P at a steady speed of v km/h where 30 ≤ v ≤ 60. Deduce (a) the time at which the 2 cars must meet, (b) the distance from P when the cars meet.

4. Two men agreed to meet at a point between their respective hometowns.

Mr A sets out at 9.30 a.m. from Ayton, driving at a steady 60 km/h, half an hour after Mr B has left Beetown which is 120 km away.

Mr B drives at 75 km/h but stops for 20 min on the way. Illustrate the men’s journeys on a displacement-time graph and use it to find the time at which the men meet and the distance of their rendezvous from Ayton.

If meet before break, about 1007 h, 36.5 km from Ayton. If meet after break, about 1018 h, 48 km from Ayton.

Distance from P (km)

Tme(h)

20

40

60

80

100

120

0900 1000 1100 1200

0

1300

u=25 u=50 v=30

v=60

Ans: (a) 1006 < t < 1111 (b) about 36 km to 76 km from P

Distance from Ayton (km)

20

60

40

80

100

120

0

0900 0930 1000 1030 Time (h)

A

B without the 20 min breaks

B with the 20 min breaks( as if start the journey 20 min later



5. A lift moves up from ground level to a 60-metre level, stops for 10 seconds and then descends to the ground in 10 seconds. The table shows the height of the lift on the upward and downward journeys, T seconds after leaving ground level:

T (s) 0 2 4 6 8 10 20 22 24 26 28 30 H (m) 0 3 16 44 57 60 60 57 44 16 3 0

(a) Taking 2 cm to represent 5 seconds on the horizontal axis, plot a graph of H (m)

against T(s) for values of T from 0 to 60 on a piece of graph paper. (b) Find the gradient of the graph at T = 8 and explains briefly what this gradient

represents. (c) A man, waiting at a 40-metre level, starts to go downstairs at T = 15. He moves at a

steady speed of 2 m/s. From the graph, find the height at which the lift passes him. 6

.

The diagram shows the two possible routes between two towns A and B. A lorry is scheduled to travel 50 km from A to B in 1 hour. After travelling 30 km, the lorry arrives at a bridge D which is too low for it to pass under. It takes the driver 5 min to turn and he then goes back along the road for 5 km until, at C, he is able to take the alternative road to B. The two roads from C to B are equal in length and apart from the turn around, the lorry travels at a speed of 50 km/h. Using a horizontal scale of 1 cm to represent 5 min and a vertical scale of 1 cm to represent 5 km from A, mark the points A and B on the vertical axis and draw the displacement-time graph for this journey. Use your graph to find how late the lorry is arriving at B. If, however, the driver had increased his average speed to v km/h as soon as he took the new road at C, he would have arrived at B at the scheduled time. Add a dotted line to your graph to show this different journey from C to B and hence estimate the value of v.

J76 II/C/8

A C B

D

25

25

205

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2 0 c m x 2 4 c m

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Worksheet 1 D-t Graphs Complete Solutions