It is a detailed analysis of speed and time of vehicles moving towards each other.
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Analogy of speed and time of vehicles moving towards each other when the vehicle's individual lengths are negligible compared to the distance covered by them: Case a): When 2 vehicles start at different time and different speed from points A and B and move towards each other: When vehicle Vlheading towards point B from A start at X A.M and vehicle V 2 heading towards A from B start at (X+a) A.M where a is in hours, and when speed of vehicles VI and V2 are respectively Sl and S2 km/hr, time taken by VI to reach the meeting point C = t hours and time taken by V2 to reach the meeting point C= (t-a) hours. If d denotes the distance between A and B in km such that AC= (d-m) km and CB= m km, .................................(2) .....................(5) 1 2 — t2/t... (Sl*t) + = d. Sl*t = d-m. S2*(t-a) = m. (3) If ti is the time taken in hours by VI to reach the remaining distance and t2 is the time taken in hours by V2 to reach the remaining distance, t2*S2 = d-m. Comparing equations (2) and (4), Sl*t = t2*S2.. Comparing equations (3) and (5), S2*(t-a) = From equation (6), From equation (7), SI/S2 = tl*Sl. (t-a)/tl.. (7) (8) Comparing equations (8) and (9), t2/t = (t-a)/tl ti*t2 = (10)
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Multiplying L.H.S of equations (8) and (9) and equating it with its R.H.S, .....................................(11) ................(13) (14) ..............(15) . .. ... ... ... .. (16) S12/S22 = (t2/t1)*((t-a)/t).. Sl/S2 = A/ (Wtl)*((t-a)/t).. (12) Example: When a carl starting from point A at 08.00 A.M heading towards point B with a speed of 60 km/hr and a car2 starting from point B at 11.00 A.M heading towards point A with a speed of 90 km/hr meets each other at 01.00 P.M, what are the time taken by carl and car2 to cover their respective remaining distances? Solution: As carl and car2 start from their respective points with a time gap of 3 hours, a = 3. They meet each other after 5 hours from the commencement of journey of carl and Hence t = 5 hours. (From equation 10) tl*t2 -10.. tl = (10/t2). • 60/90 = ((t2/ ... .... (From equations 12 and 13) Solving, we get, t2 = (10/3) hours and ti = 3 hours. Hence carl and car 2 take 3 and (10/3) hours respectively to cover their remaining distances after their meeting point. Case b): When 2 vehicles start at same time and different speed from points A and B and move towards each other: In this case, the time difference 'a' becomes zero. Hence equation (10) becomes, ti*t2 = Equation (12) becomes, Sl/S2 = (t2/tl) Example:When a bikel starting from point A heading towards point B travelling at a speed of 40 km/hr and bike2 starting from point B heading towards point A travelling at a speed of 70 km/hr meet each other after 6 hours, what are the time taken by bike 1 and 2 to cover their remaining distances if they start their journey simultaneously? Solution: Here, ti 36. = (36/t2). • . (From equation 14)
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40/70 = (W (36/t2)) ..... .... ..... .... ..... (From equations 15 and 16). Solving, we get, ti = (21/2) hours and t2 = (24/7) hours Hence bikel and bike 2 take (21/2) and (24/7) hours respectively to cover their remaining distances after their meeting point. Case c): When 2 vehicles start at different time with same speed from points A and B and move towards each other: ...........................(17) .. ..... ..... ...................... (18) ................(19) ..... ..... ... (20) . .... ... ... ... ... ... ... .... (21) In this case, Sl = S2. Hence equation (1) becomes, (Sl*t) + = d Till the meeting point, Sl*t = d-m. After the meeting point, ti*S1 = m.. t2*S1 = d-m. Comparing equations (19) and (20), t = ti+a... Comparing equations (18) and (21), t2 ..........................(22) (23) Example : When a mopedl starting from point A at 07.00 A.M heading towards point B and a moped 2 starting from point B at 11.00 A.M heading towards point A both with a speed of 50 km/hr meets each other at 01.00 P.M a) What are the time taken by mopedl and moped2 to cover their respective remaining distances? b) What is the distance between point A and B? Solution: a) Here, a = 4 hours. 6 = ti+4 (From equation 22)
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Hence ti = 2 hours t2 = t = 6 hours (From equation 23) Hence mopedl and moped 2 take 2 and 6 hours respectively to cover their remaining distances after their meeting point. b) d = = 400 km (From equation 17). Hence distance between point A and point B is 400 km. Case d): When 2 vehicles start at same time with same speed from points A and B and move towards each other: In this case, d-m = m and Sl = S2 m d/2. ... ... ... ... .... (24) .........................(25) (26) Example: When a autol starting from point A heading towards point B and aut02 starting from point B heading towards point A both travelling at a speed of 60 km/hr meet each other after 6 hours, a) What are the time taken by auto 1 and 2 to cover their remaining distances if they start their journey simultaneously? b) What is the distance between point A and B? c) What is the distance from A to the meeting point? Solution: = t2 = t = 6 hours (From equation 26) Hence autol and auto 2 both take 6hours to cover their remaining distances after their meeting point. b) d = = 720 km (From equation 25) Hence distance between point A and point B is 720 km. c) m = 720/2 =360 km (From equation 24) Hence distance from point A to the meeting point is 360 km.
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