27
Problems about the motion of two bodies
Textbook: Part 1, lesson 27, pp. 106–109
GoalSolve problems on the motion of two bodies (moving apart, towards each other, chasing) using the speed of approach or separation.
New words
speed of separation · uzoqlashish tezligispeed of approach · yaqinlashish tezligiopposite directions · qarama-qarshi yo‘nalishmeeting · uchrashish
Explanation
If two bodies start from one point in opposite directions, the distance between them grows by v₁ + v₂ every hour; this is the speed of separation. If they move towards each other, the distance shrinks by v₁ + v₂ every hour; this is the speed of approach, and the meeting time is t = S : (v₁ + v₂). If they move in the same direction and the one behind is faster, the gap shrinks by v₁ − v₂ every hour. Divide the gap by that speed to find the time to catch up. A drawing or table helps to understand the problem.
Worked examples
Two cyclists leave one place in opposite directions at 15 km/h and 12 km/h. After 3 hours they are (15 + 12) × 3 = 81 km apart.
Walkers set out towards each other from villages 36 km apart at 4 km/h and 5 km/h: approach speed 9 km/h, they meet after 36 : 9 = 4 hours. Chase: gap 28 km, speeds 15 and 8 km/h, so 28 : (15 − 8) = 4 hours.
Class activity
“Two pupils — one aisle”: two pupils walk towards each other from the ends of the aisle at the same pace; the class predicts beforehand how many steps are needed before they meet (safely, slowly).
Practice
1
Two boats leave a shore in opposite directions at 30 km/h and 40 km/h for 2 hours. How many km apart are they?
140
2
Two children 270 m apart walk towards each other at 40 m/min and 50 m/min. After how many minutes do they meet?
3
3
The gap is 28 km. The cyclist behind goes 15 km/h and the pedestrian ahead 8 km/h in the same direction. After how many hours does the cyclist catch up?
4
4
Why are speeds added for opposite directions and subtracted for the same direction?
In opposite directions both bodies change the distance together; in the same direction the front body adds to the gap and the back body closes it.