Solving Simultaneous Linear Equations on the Problems of Relative Motion.
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Transcript of Solving Simultaneous Linear Equations on the Problems of Relative Motion.
Solving Simultaneous Linear Solving Simultaneous Linear EquationsEquations
on the Problems of on the Problems of
Relative Motion
Two cars A and B are 140km apart
A B140km
Basic term
Basic term
They travel towards each other
A B
They meet !
Basic term
They travel in the same direction
A B
Car A catches up with Car B
Speed
The speed of a car is 50 km/h.
The speed of a car is 50 km in one hour.
The speed of another car is 100 m/min.
The speed of another car is 100 m in one minute.
Speed Formula: Time
DistanceSpeed
Distance = Speed × Time
Learn how to set up equations
to solve the problems
Question 1 A and B are 21 km apartThey walk towards each other They will meet after 3 hours
What are the speeds of A and B?
A B21 km
They meet after 3 hours: x km
: y km
Let x be A’s speed and y be B’s speed.
After 3 hours, how far will A walk ?
How to equate the distances? 3x + 3y = 21
3x km
3y kmAfter 3 hours, how far will B walk ?
km/h km/h
3x km 3y km
Question 1
A and B are 21 km apart
Walking towards each other
Set up an equation with
2 unknown speeds
A B18 m
A will catch up with B after 4 minutes.
Question 2
• A and B are 18 m apart.
• Walking in the same direction
Set up an equation with 2 unknown speeds.
Choice A
A B18 m
A will catch up with B after 4 minutes.
Question 2
• A and B are 18 m apart.
• Walking in the same direction
Set up an equation with 2 unknown speeds.
Choice B
A B18 m
A will catch up with B after 4 mins. : x m
: y m
Let x m/min be A’s speed and y m/min be B’s speed.
How far will A walk after 4 minutes? 4x m
How far will B walk after 4 minutes? 4y m
4x m
4y m
How to equate the distances? 4x – 4y = 18
Question 2
• A and B are 18 m apart.
• Walking in the same direction
Set up an equation with 2 unknown speeds.
Variables
the speeds
the distance apart
the time
Question 3
A car and a bicycle are a certain distance apart.
Speed of the car : 65km/h
Traveling towards each other, they meet in 2 hours.
the speed of bicycle
the distance apart
Two unknowns:
-- x km
-- y km/h
Do worksheet : Q.3
(a) Draw a diagram to show the situation.
(b) Set up an equation with the unknown distance and speed.
Question 3Speed of the car : 65km/h
Let x km be the distance apart and y km/h be the speed of the bicycle.
Car Bicyclex km
(65 2) km 2y km
65 2 + 2y = x
They meet after 2 hours
Traveling towards each other, they meet after 2 hours
Question 4
Traveling in the same direction,
train N will catch up with train M in 2.5 hours.
Speed of train N : 152 km/h
Two trains M and N are a certain distance apart.
Do worksheet : Q.4
(a) Draw a diagram to show the situation.
(b) Set up an equation with the unknown speed and time.
Question 4
2 trains are a certain distance apart. Speed of train N : 152km/h
Let x km be the distance apart and y km/h be the speed of train M.
Train N Train M
x km
After 2.5 hours152 2.5 km
152 2.5 – 2.5y = x
2.5y km
Train N will catch up with train M in 2.5 hours.
480 m
Towards each other
Question 5
480 m
After 1 minTowards each other
480 m
After 2 minsTowards each other
480 m
Meet in 3 minsTowards each other
480 m
Same direction
480 m
Meet in 3 minsTowards each other
480 m
Same direction After 2 mins
480 m
Meet in 3 minsTowards each other
480 m
Same direction After 4 mins
480 m
Meet in 3 minsTowards each other
480 m
Same direction After 6 mins
480 m
Meet in 3 minsTowards each other
480 m
Same direction The dog catches up with the cat in 8 mins
480 m
Meet in 3 minsTowards each other
Their speeds??
Question 5
A dog and a cat are 480 m apart.
3x + 3y = 480
8x – 8y = 480
Let x m/min be the speed of the dog and y m/min be the speed of the cat.
3y m 3x m
480 m
8x m
8y m480 m
Traveling towards each other, they will meet in 3 minutes.
Traveling in the same direction,the dog will catch up with the cat in 8 mins.
Solve the simultaneous linear equations:
(1)
(2)
The speed of the dog is 110 m/min and the speed of the cat is 50 m/min.
Do worksheet : Q. 6
3y m 3x m
480 m
8x m
8y m480 m
From (1), 3(x + y) = 480 x + y = 160 (3)
From (2), 8(x – y) = 480 x – y = 60 (4)
(3) + (4): 2x = 220 x = 110
(3) – (4): 2y = 100 y = 50
3x + 3y = 480
8x – 8y = 480
Question 6
Ann and Teddy are 60 km apart.
Teddy Ann60 km
1.5y km 1.5x km 1.5x +1.5 y = 60
4y km
4x km
4y – 4x = 60
Cycling towards each other, they will meet in 1.5 hours.
Cycling in the same direction, Teddy will catch up with Ann in 4 hours.
Teddy Ann60 km
Let x km/h be the speed of Ann’s bicycle and y km/h be the speed of Teddy’s bicycle.
Their speeds ??
Solve the simultaneous linear equations:
The speed of Ann’s bicycle is 12.5 km/h and the speed of Teddy’s bicycle is 27.5 km/h.
(1)
(2)
From (1), 1.5(x + y) = 60 x + y = 40 (3)From (2), 4(y – x) = 60 y – x = 15 (4)
(3) + (4): 2y = 55 y = 27.5
(3) – (4): 2x = 25 x = 12.5
1.5x + 1.5y = 60
4y – 4x = 60
Harder Problem 1Harder Problem 1
Kenneth and Betty are 200 km apart.
If driving towards each other, they meet in 2 hours.
If Betty starts driving at noon, andKenneth starts in the same direction at 1 p.m., Kenneth will catch up with Betty at 6:45 p.m.
Set up two equations with two unknown speeds.
Harder ProblemKenneth and Betty are 200 km apart.If driving towards each other, they meet in 2 hours. If Betty starts driving at noon, and Kenneth starts in the same direction at 1 p.m., Kenneth will catch up with Betty at 6:45 p.m. Set up two equations with two unknown speeds.
Let x km/h be the speed of Kenneth’s car and y km/h be the speed of Betty’s bicycle.
2x + 2y = 200
20060
456
60
455 yx
2004
36
4
35 yxor
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. Set up two equations with two unknown speeds.
Harder Problem 2 Harder Problem 2 (Circular motion)(Circular motion)
Quick review
• The key in setting up equations to solve problems of relative motion:
Equate the distances !
Use of Theory of Learning and Variation
變易理論的運用變 Variant 背景 Background
(Invariant)辨識特徵
Critical feature
Relative Motion (Linear and Circular)
Moving towards each other (meeting problem)
Speed, distance, or timein different units
Speed formula:
Distance = Speed × Time
Using the speed formula to equate the distances traveled to set up equations
Moving in the same direction (catch-up problem)
Both meeting and catch-up problems
http://www.sttss.edu.hk/Mathematics/