solving simultaneous linear equations on the problems of linear relative motion
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Solving simultaneous linear equations on the problems of linear relative motion. Speed Formula:. Distance = Speed × Time. - PowerPoint PPT PresentationTRANSCRIPT
Solving simultaneous linear equations
on the problems of
linear relative motion
Speed Formula: Time
DistanceSpeed Distance = Speed × Time
e.g.1 ) Two cars A and B are at a certain distance apart. The speed of car A is 72 km/h while the speed of car B is 48 km/h. If they start at the same time and they travel towards each other, they will meet in two hours. Find the distance between them.
A B
They meet in two hours
The distance between them : 144 km + 96 km = 240 km
72 x 2 = 144 km 48 x 2 = 96 km
e.g.2) May and Bobby are at a certain distance apart. The walking speed of May is 3km/h and that of Bobby is 7 km/h. If they walk in the same direction, Bobby will catch up with May in 5 hours.
Find the distance between them.
The distance between them : 35 km - 15 km = 20 km
7 x 5 = 35 km
3 x 5 = 15 km
Bobby May
Learn how to set up equations
to solve the problems
A B42 km
They meet after 3 hours : x km
: y km
Let x be A’s speed and y be B’s speed
After 1 hour, how far will A walk ?
After 3 hours, how far will A walk ?
How to equate the distances ?
3x + 3y = 42
x km
y kmAfter 1 hour, how far will B walk ?
3x km
After 3 hours, how far will B walk ? 3y km
e.g.3 ) A and B are 42 km apart. If they walk towards each other, they will meet after 3 hours. Set up an equation with two unknown speeds.
km/h km/h
A B22 km
A will catch up with B after 9 hours : x km
: y km
Let x km/h be A’s speed and y km/h be B’s speed
How far will A walk after 1 hour ? x km
How far will A walk after 9 hours ? 9x km
How far will B walk after 1 hour ? y km
How far will B walk after 9 hours ? 9y km
9x km
9y km
How to equate the distances ?
or 9x = 22 + 9y 9x – 9y = 22
e.g. 4) A and B are 22 km apart. If they walk in the same direction, A will catch up with B after 9 hours. Set up an equation with two unknown speeds.
Do worksheet : No.1-2
1) Two trains M and N are 250 km apart. If they start at the same time and they travel towards each other, they will meet after 50 minutes. Set up an equation with two unknown speeds.
Let x km/min be the speed of train M and y km/min be the speed of train N.
Train M Train N250 km
After 50 minutes50x km 50y km
50x + 50y = 250
Let x km/h be the speed of train M and y km/h be the speed of train N.
250y60
50x
60
50
2) Jacky and Amy are 60 km apart. Jacky takes a minibus. Amy travels by her car in the same direction as the minibus and overtakes it after 7 hours. Set up an equation with two unknown speeds.
Let x km/h be the speed of the minibus and y km/h be the speed of Amy’s car.
minibusAmy’s car60 km
After 7 hours7y km
7x km
7y – 7x = 60 or 7y = 60 + 7xDo worksheet : No. 3,4
3) Tommy and Martin ride bicycles on the same road at constant speeds and they are a certain distance apart. The speed of Martin’s bicycle is 15 km/h. If they travel in the same direction, Tommy’s bicycle will catch up with Martin’s bicycle in 8 hours. a) Draw a diagram to show the situation. b) Set up an equation with the unknown distance apart and the unknown speed of Tommy’s bicycle.
Let x km be the distance apart and y km/h be the speed of Tommy’sbicycle.
Tommy’s bicycle Martin’s bicyclex km
After 8 hours8y km
8y – 120 = x or 8y = x + 120
158 = 120 km
4) A car and a bicycle are 72 km apart. The speed of the bicycle is 12 km/h. If they travel towards each other, they will meet after some time. a) Draw a diagram to show the situation. b) Set up an equation with the unknown time and the unknown speed of the car.
Let x hours be the time and y km/h be the speed of the car.
car bicycle72 km
xy km 12x km
xy + 12x = 72
They meet after x hours
e.g.5) Two cars P and Q are 480 km apart. If they start at the same time and travel towards each other, they will meet in three hours. If they travel in the same direction, car Q will overtake car P in eight hours. Find the speeds of cars P and Q.
P Q480 km
3x km 3y km3x + 3y = 480
P Q480 km
8y km
8x km
8y – 8x = 480 or 8y = 8x + 480
Let x km/h be the speed of car P and y km/h be the speed of car Q.
48088
48033
xy
yx
Solve the simultaneous linear equations:
…(1)
…(2)
The speed of car P is 50 km/h and the speed of car Q is 110 km/h.
)4...(14402424:3)2(
)3...(38402424:8)1(
xy
yx
50
240048:)4()3(
x
x
Substitute into (1),50x
110
3303
4803150
48033
y
y
y
yx
Do worksheet : No. 5
5) Teddy and Ann are a certain distance apart. They ride bicycles at uniform speeds. The speed of Teddy’s bicycle is 18 km/h. If they ride towards each other, they will meet in 2 hours. If they ride in the same direction, Teddy will overtake Ann in 10 hours. Find the speed of Ann’s bicycle and the original distance apart. ( Set up two simultaneous linear equations.)
Let x km/h be the speed of Ann’s bicycle and y km be the original distance apart.
Teddy’s bicycle Ann’s bicycley km
36 km 2x km 36 + 2x = y
Teddy’s bicycle Ann’s bicycley km
180 km
10x km
180 – 10x = y
182
1810
yx
yx
10180
236
Solve the simultaneous linear equations:
…(1)
…(2)
Substitute (1) into (2), Substitute x = 12 into (1), y + 10x = 180 36 + 2x = y36 + 2x + 10x = 180 36 + 24 = y 12x = 180 – 36 y = 60 12x = 144 x = 12
The speed of Ann’s bicycle is 12 km/h and the original distance is 60 km.
Solving simultaneous linear equations
on the problems of
circular relative motion
e.g.6) Cat A and cat B are running around a 640m circular track. Cat A runs faster. If they start together ( at the same time and position ) and they go in opposite directions, they will meet in 35 seconds later.
AB
35 seconds later
35x m
35y m
35x + 35y = 640
Let x m/s be cat A’s speed and y m/s be cat B’s speed . Can you draw the paths run by cats A and B ?
How far does cat A run in terms of x?
How to equate the distances ?
e.g.7) Cat A and cat B are running around a 640m circular track. Cat A runs faster. If they start together ( at the same time and position ) and they go in the same direction, cat A will catch up with cat B in 1 minute and 15 seconds later.
A
B
1 minute and 15 seconds later
75x m
75y m
How to equate the distances ?
or 75x = 75y + 640
75x – 75y = 640
Do worksheet : No.6,7
Let x m/s be dog A’s speed and y m/s be dog B’s speed .
How far does A run in terms of x?
6) Sammy and Judy are practicing on a 600m circular track. Sammy runs faster than Judy.If they start together ( at the same time and position ) and they go in opposite directions, they will meet 40 seconds later.
Let x m/s be Sammy’s speed and y m/s be Judy’s speed . Set up an equation with x and y.
Sammy Judy
After 40 seconds
40y km 40x km
40x + 40y = 600
7) In the sports day, Kenneth and Sally join the 1500m running race and run on a 400m circular track. If they start together, Kenneth will overtake Sally 5 minutes later. Let x m/min be Kenneth’s speed and y m/min be Sally’s speed . Set up an equation with x and y.
Kenneth
Judy
5 minutes later
5x m5y m or 5x = 5y + 400
5x –5y = 400
e.g.8) Susan and Peter are running on a 900m circular track outside the playground. Peter runs faster than Susan. If they start together and run in the same direction, Peter will catch up with Susan 6 minutes later. If they go in opposite directions, they will meet 1.2 minutes later. Find their speeds. Let x m/min be Susan’s speed and y m/min be Peter’s speed . Peter
SusanPeterSusan
6 minutes later1.2 minutes later
6y m
6x m1.2x m
1.2y m
6y – 6x = 900 or 6y = 6x + 900 1.2x + 1.2y = 900
9002.12.1
90066
yx
xy … (1)
… (2)
)3...(450066:5)2( yx
450
540012:)3()1(
y
y
Substitute into (2),450y
300
3602.1
9005402.1
9002.12.1
x
x
x
yx
Susan’s speed is 300 m/min and Peter’s speed is 450 m/min.
Do worksheet : No.8
8) James and Ken are jogging round a circular park. Ken jogs faster. If they start together and jog in opposite directions, they will meet 50 seconds later. If they go in the same direction, Ken will overtake James 2.5 minutes later. If James’ jogging speed is 3m/s, find the jogging speed of Ken and the length of the circular park.
Let x m/s be Ken’s speed and y m be the lengthlength of the circular park.
KenJames
KenJames
50 seconds later
50x m= 150m
50x + 150 = y
503
2.5 minutes later
150x m
= 450m1503
150x = 450 + y or 150x – 450 = y
yx
yx
450150
15050 …(1)
…(2)
Substitute (1) into (2),
6
600100
15050450150
450150
x
x
xx
yx
6xSubstitute into (1),
450
150300
15050
y
y
yx
Ken’s speed is 6 m/s and the length of the circular park is 450m.
There are two people running on a circular track.Write an equation to relate the distances travelled by the two persons
for the nth catch-up on the circular track.
Let x m be the distance travelled by the faster one, y m be the distance travelled by the slower one and z m be the circular track length.
Harder Problem:
Four Types of Relative Motion
What is the critical feature in setting up equations to solve these relative motion problems?