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CHAPTER 6 : DISTANCE AND AREA UNDER A GRAPH NOTES : 1. Distance-Time graph Distance

a) Speed = Gradient of a distance- time graph = Change of distance = y Change of time x b) Average speed = Total distance travelled Total time taken 2. Speed - Time Graph

a) Acceleration or deceleration(Rate of change of speed) = Gradient of a speed- time graph = Change in speed Change in time b) Total distance travelled = Area under the speed –time graph

0

stationary

y x

Time

uniform speed

constant speed

Area = Total distasnce travelled

acceleration

deceleration

0 Time

Speed

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Exercise 1 1. The diagram shows the distance-time graph of a particle for a period of 30 s. State the length of time in which the particle is stationary. Answer: 2. The diagram shows the distance-time graph of a particle for a period of 4 s. Find the distance from P to Q. Answer:

0

0

Q

P

Time (s) 30 1510

700

500

Distance (m)

Time (s) 4

30

5

Distance (m)

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3. The distance-time graph above shows the journey of a van over a period of 6 hours. Find the speed, in km h-1, of the van in the first 2 hours. Answer: 4. The distance-time graph above illustrates the journey of a car over a period of 4 hours. Find the average speed, in km h-1, of the car for the whole journey. Answer:

Time (hours) 1.5 21

0Time (hours)

6 32

160

Distance (m)

0 4

100

Distance (km)

60

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5. The diagram shows the distance-time graph of a particle for a period of 40 minutes. Find the average speed from P to Q. Answer: 6. The diagram shows the speed-time graph of a particle for a period of 9 seconds. Find the length of time at constant speed. Answer:

Q

P0 Time (minutes) 40 15

36

20

Distance (km)

0 Time (s) 9 5

12

8

Speed (ms-1)

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7. The diagram shows the speed-time graph of a particle for a period of 7 seconds. Find the rate of change of speed in the last 4 s. Answer: 8. The diagram shows the speed-time graph of a particle for a period of 9 seconds. Find the distance travelled in the last 5 s. Answer:

0

0 Time (s) 7 3

20

Speed (ms-1)

Time (s) 9 4

10

Speed (ms-1)

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9. The diagram shows the speed-time graph of a particle for a period of 10 seconds. Find the total distance travelled in the first 7 s. Answer: 10. The diagram shows the speed-time graph of a particle for a period of 10 seconds. Find the rate of change of speed in the last 3 s. Answer:

0

4

0 Time (s) 10 75

16

Speed (ms-1)

Time (s) 10 73

18

13

Speed (ms-1)

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Exercise 2 1. The figure shows the distance-time graph of Mr. Tan’s journey from town P to town Q and then to town R. Calculate a) the speed from town P to town Q,

b) the speed from town Q to town R,

c) the average speed for the whole journey.

2. The figure shows the distance-time graph of an object travelling from town P to town S. Calculate a) the speed from P to Q,

b) the speed from Q to R, c) the speed from R to S.

Distance (km)

S

R Q

P

1000

Q

R

0 P Time (h) 5 2

200

125

Distance (km)

Time 1430 12001100

80

50

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3. The figure shows the speed-time graph of an object during a period of 10 seconds. Calculate a) the speed of the object when t = 3 b) the distance travelled in the first 10

seconds.

4. The figure shows the speed-time graph of an object for a particular journey. Calculate a) the acceleration during the first 10 seconds,

b) the total distance travelled for the whole journey.

0

0

24

10 6

100

Time (s)

Speed (ms-1)

Time (s) 70 3010

9

Speed (ms-1)

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5. The figure shows the speed-time graph of a bicycle and a bus during a period of 90 seconds. Calculate a) the distance travelled by the bicycle during the 90 seconds,

b) the distance travelled by the bus during the 90 seconds.

6. The figure shows the distance-time graph of a particle over a period of 12 seconds. Calculate a) the speed of the particle, from 5 seconds to 9 seconds,

b) the average speed of the particle, for the whole journey.

0 12

0

bicycle

bus

4

Time (s) 90 7030

6

Speed (ms-1)

Time (s) 9 52

70

40

Distance (m)

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7. The figure shows the distance-time graph of an object over a period of 14 seconds .The speed of the object during the first 6 seconds is 3 ms-1. Calculate a) the value of y, b) the average speed of the object for the

whole journey.

8. The figure shows the speed-time graph of a particle for a period of 10 seconds. Calculate the rate of change of speed, in m s-2, during the first 4 seconds. Answer:

0 Time (s) 14 6

y

4

Distance (m)

Speed(m s-1 )

0 Time (s) 10 4

12

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9. The figure shows the speed-time graph of a particle during a period of t seconds a) Calculate the rate of change of speed during the first 4 seconds.

b) Given that the total distance travelled in the t seconds is 124m, find the value of t.

10. The figure shows the speed-time graph of a train over a period of 90 seconds. Calculate a) the deceleration of the train during the first 20 seconds,

b) the total distance travelled during the 90 seconds.

0

0Time (s) t4

20

12

Speed (m s-1)

Time (s) 90 65

15

9

Speed (ms-1)

20

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Diagnostic Test

Distance(m) 1. 12 u

Time (s) 0 3 8 The diagram shows a distance- time graph of an object over a period of 8 seconds. Given that its uniform speed in the first 3 seconds is 2 ms 1− , calculate (a) the value of u, (b) the distance travelled in the last 5 seconds. (c) its average speed during the 8 seconds.

Speed (ms 1− ) 2. u 16 0 4 12 20 Time (s) The diagram shows the speed-time graph of an object over a period of 20 seconds. Calculate

(a) the rate of change of speed in the first 4 seconds, (b) the value of u, if the distance travelled in the last 8 seconds is 160m.

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3. Speed (ms 1− ) 12 u Time(s) 0 5 15 The diagram shows a speed –time graph of an object over a period of 15 seconds. Calculate (a) the distance travelled at uniform speed,

(b) the value of u, if its average speed during the first 15 seconds is 11 ms 1− ]

4. Speed (ms 1− ) 20 5 0 16 t Time (s) The diagram shows speed-time graph of an object over a period of t seconds. Given that the total distance travelled by the object during the t seconds is 115 meter, calculate (a) the distanvce travelled from the 16th seconds to the t-seconds, (b) the value of t.

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5. Speed (ms 1− ) v 2 0 4 10 Time (s) The diagram shows the speed-time graph of an object over a period of 10 seconds. Calculate the value of v for each situations below : (a) if the rate of change of speed during the first 4 seconds is -3 ms 2− . (b) if the total distance travelled during the 10 seconds is 28 meter.

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CHAPTER 6 : GRADIENT AND AREA UNDER A GRAPH Answers Exercise 1

1. 5 s 2. 25 m 3. 80 4. 50 5. 0.9 km / min 6. 5 s 7. -5 ms-2 8. 50 m 9. 72 m

10. 35 ms-2

Exercise 2 1a) 62.5 kmh-1 b) 25 kmh-1 c) 40 kmh-1 2a) 30 kmh-1 b) 0 kmh-1 c) 20 kmh-1 3a) 62 b) 468 m 4a) 0.9 ms-2 b) 405 m 5a) 360m b) 390 m

6a) 7.5 ms-1 b) 1132 ms-1

7a) 22 b) 2 76 ms-1

8) 3 9a) 2 m s-2 b) 7 10a) 0.3 ms-2 b) 945m

Diagnostic Test 1. (a) 6 (b) 6 (c) 1.5 2. (a) 4 (b) 40 3. (a) 120 (b) 6 4. (a) 75 (b) 22 5. (a) 14 (b) 9