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Solving simultaneous linear equations on the problems of linear relative motion

Solving simultaneous linear equations on the problems of linear relative motion. Speed Formula:. Distance = Speed × Time.

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Solving simultaneous linear equations on the problems of linear relative motion

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  1. Solving simultaneous linear equations on the problems of linear relative motion Speed Formula: Distance = Speed × Time

  2. 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 72 x 2 = 144 km 48 x 2 = 96 km The distance between them : 144 km + 96 km = 240 km

  3. 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 upwith May in 5 hours. Find the distance between them. 7 x 5 = 35 km 3 x 5 = 15 km Bobby May The distance between them : 35 km - 15 km = 20 km

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

  5. km/h km/h : x km : y km A 42 km B 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. Let x be A’s speed and y be B’s speed They meet after 3 hours After 1 hour, how far will A walk ? x km After 3 hours, how far will A walk ? 3x km After 1 hour, how far will B walk ? y km After 3 hours, how far will B walk ? 3y km 3x + 3y = 42 How to equate the distances ?

  6. : x km : y km B A 22 km 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. Let x km/h be A’s speed and y km/h be B’s speed A will catch up with B after 9 hours 9x km 9y km 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 How to equate the distances ? 9x – 9y = 22 or 9x = 22 + 9y Do worksheet : No.1-2

  7. Train M Train N 250 km 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. After 50 minutes 50x km 50y km Let x km/h be the speed of train M and y km/h be the speed of train N. 50x + 50y = 250

  8. 60 km 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. After 7 hours 7y km 7x km Amy’s car minibus 7y – 7x = 60 or 7y = 60 + 7x Do worksheet : No. 3,4

  9. 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 distanceapart and y km/h be the speed of Tommy’s bicycle. After 8 hours 8y km = 120 km x km Tommy’s bicycle Martin’s bicycle 8y – 120 = x or 8y = x + 120

  10. bicycle car 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 thetime and y km/h be the speed of the car. They meet after x hours 12x km xy km 72 km xy + 12x = 72

  11. 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 P 480 km 480 km Q Q   Let x km/h be the speed of car P and y km/h be the speed of car Q. 3y km 3x + 3y = 480 3x km 8y km 8x km 8y – 8x = 480 or 8y = 8x + 480

  12. Substitute into (1), Solve the simultaneous linear equations: …(1) Do worksheet : No. 5 …(2) The speed of car P is 50 km/h and the speed of car Q is 110 km/h.

  13. 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.) Teddy’s bicycle Ann’s bicycle y km   Let x km/h be the speed of Ann’s bicycle and y km be the original distance apart. 36 km 2x km 36 + 2x = y y km 180 – 10x = y 180 km 10x km Teddy’s bicycle Ann’s bicycle

  14. Solve the simultaneous linear equations: …(1) …(2) Substitute (1) into (2), Substitute x = 12 into (1), y + 10x = 180 36 + 2x = y 36 + 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.

  15. Solving simultaneous linear equations on the problems of circular relative motion

  16. 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. A B 35 seconds later 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 ? 35x m How far does cat A run in terms of x? 35y m How to equate the distances ? 35x + 35y = 640

  17. 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 withcat B in 1 minute and 15 seconds later. A B 1 minute and 15 seconds later 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? 75x m How to equate the distances ? 75x – 75y = 640 75y m or 75x = 75y + 640 Do worksheet : No.6,7

  18. After 40 seconds 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. 40x + 40y = 600 Sammy Judy 40y km 40x km

  19. 5 minutes later 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 5x –5y = 400 5x m or 5x = 5y + 400 5y m

  20. 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. Peter Susan Peter Susan 1.2 minutes later 6 minutes later Let x m/min be Susan’s speed and y m/min be Peter’s speed . 1.2x m 6x m 6y m 1.2y m 6y – 6x = 900 or 6y = 6x + 900 1.2x + 1.2y = 900

  21. Substitute into (2), … (1) … (2) Do worksheet : No.8 Susan’s speed is 300 m/min and Peter’s speed is 450 m/min.

  22. 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. Ken James Ken James 50 seconds later 2.5 minutes later Let x m/s be Ken’s speed and y m be the length of the circular park. 150x m 50x m = 150m = 450m 50x + 150 = y 150x = 450 + y or 150x – 450 = y

  23. Substitute into (1), …(1) …(2) Substitute (1) into (2), Ken’s speed is 6 m/s and the length of the circular park is 450m.

  24. Harder Problem: 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.

  25. Four Types of Relative Motion What is the critical feature in setting up equations to solve these relative motion problems?

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