2. Solving Simultaneous Linear Equations on the Problems of Relative Motion

3. Two cars A and B are 140km apart A B 140km Basic term

4. They travel towards each other A B They meet !

5. Basic term They travel in the same direction A B Car A catches up with Car B

6. 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: Distance = Speed × Time

7. Learn how to set up equations to solve the problems

8. Question 1  A and B are 21 km apart  They walk towards each other  They will meet after 3 hours What are the speeds of A and B?

9. AB21 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 3x km3y km Question 1 A and B are 21 km apart Walking towards each other Set up an equation with 2 unknown speeds

10. A B 18 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

11. A B 18 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

12. A B 18 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.

13. Variables  the speeds  the distance apart  the time

14. 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

15. (a)Draw a diagram to show the situation. (b) Set up an equation with the unknown distance and speed. Question 3 Speed of the car : 65km/h Let x km be the distance apart and y km/h be the speed of the bicycle. Car Bicycle x km (65  2) km 2y km 65  2 + 2y = x They meet after 2 hours Traveling towards each other, they meet after 2 hours

16. 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

17. (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 hours 152  2.5 km 152  2.5 – 2.5y = x 2.5y km Train N will catch up with train M in 2.5 hours.

18. 480 m Towards each other Question 5

19. 480 m After 1 min Towards each other

20. 480 m After 2 mins Towards each other

21. 480 m Meet in 3 mins Towards each other

22. 480 m Same direction 480 m Meet in 3 mins Towards each other

23. 480 m Same direction After 2 mins 480 m Meet in 3 mins Towards each other

24. 480 m Same direction After 4 mins 480 m Meet in 3 mins Towards each other

25. 480 m Same direction After 6 mins 480 m Meet in 3 mins Towards each other

26. 480 m Same direction The dog catches up with the cat in 8 mins 480 m Meet in 3 mins Towards each other Their speeds??

27. 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 m 480 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. 

28. 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 m3x m 480 m 8x m 8y m 480 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

29. Question 6 Ann and Teddy are 60 km apart. TeddyAnn 60 km 1.5y km1.5x 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.   TeddyAnn 60 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 ??

30. 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

31. Harder Problem 1 Harder 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, 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.

32. Harder Problem Kenneth 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 or

33. 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 (Circular motion)

34. Quick review The key in setting up equations to solve problems of relative motion: Equate the distances !

35. 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 time in 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/