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| *Vultures Knob>>>Road Bike |
Four Vehicles? |
A car, a van, a truck and a bike are all travelling in the same direction on the same road, each at its own constant speed. At 10am, the car overtakes the van, at noon, it overtakes the truck, at 2pm it overtakes the bike. At 4pm the truck overtakes the bike and at 6pm, the van overtakes the truck. a) The speed of the car is 120km per hour and the speed of the truck is 80km/h. Find the speeds of the van and the bike and also the time at which the van overtakes the bike. b) Let c and T represent the speeds in km/h of the car and the truck respectivelty. Find the speeds of the van and the bike in terms of c and T. Show that the time when the van overtakes the bike is the same. regardless of the speeds of the car and the truck. Let the speeds of van, car, truck and bike be 鈥榲鈥? 鈥榗鈥? 鈥榯鈥?and 鈥榖鈥?respectively. Find the fig. in the attachment to the mail I sent you. Distance travelled by van in 8 hrs (10 am to 6 pm) = V10V18 = C10C12 + T12T18 = 2c + 6t = 2(c + 3t) km. Therefore speed of the van, v = 2(c + 3t)/8 = (c + 3t)/4 km/hr ____________(1) Distance travelled by bike in 2 hrs (2 pm to 4 pm) = B14B16 = C14T16 = T12T16 鈥?C12C14 = 4t 鈥?2c = 2(2t 鈥?c) km. Therefore speed of the bike, b = 2(2t 鈥?c)/2 = (2t 鈥?c) km/hr _____________(2) PART (b) Let the van overtake the bike after 鈥榯0鈥?hours from 10 am. The condition for this overtake is: Distance travelled by van in 鈥榯0鈥?= Distance travelled by bike in 鈥榯0鈥?hrs + Distance by which the van was behind the bike at 10 am Now, V10B10 = C10C14 鈥?B10B14 = 4c 鈥?4b = 4(c 鈥?b) = 4(c 鈥?2t + c) = 8(c 鈥?t) as b = 2t 鈥?c. So, v3t0 = b3t0 + V10B10 = b3t0 + 8(c 鈥?t) or t0 = 8(c 鈥?t)/(v 鈥?b) _____________(3) Also, v 鈥?b = (c + 3t)/4 鈥?(2t 鈥?c) = 5(c 鈥?t)/4 ____________(4) Substituting the value of (v 鈥?b) from (4) in (3) we get, t0 = 32/5 hours i.e. fixed and independent of c, t. PART (a) Given c = 120 km/hr, t = 80 km/hr. From (1) v = (120 + 3380)/4 = 90 km/hr and from (2) b = 2380 鈥?120 = 40 km/hr. However, t0 remains the same and so is equal to 6 hr 24 mins. fgktrh I think your question is inaccurate. If the the van OVERTAKES the truck, then it is only logical that the speed of the van is faster than the truck's. But caculations show that the van, in order to be at the same point as the truck at 6pm, the van has to travel at a speed slower speed than the truck. u r confusing me Ok, here we go! 1) At noon the car and the truck are level and since the car is travelling 40km/h faster than the truck, this means the truck must have been 80km ahead of the car at 10am. 2) At 2pm, the car will have travelled a total or 480km and be level with the bike. The truck will be 80km behind the car and since the car and bike are level, this means that the truck is also 80km behind the bike. 3) At 4pm, the truck is level with the bike. In other words it's taken the truck 2 hours to make up the 80km that it was behind at 2pm. Since we know that the truck travels at 80km/h it means that the truck will have travelled 160km in those 2 hours. This means that the bike must have travelled 80km in those same 2 hours in order for them to be level by 4pm. SO THE BIKE'S SPEED IS 40km/h. (1st answer). 4) At 10am the car and van are level and the truck is 80km ahead (see (1) above). Since the van doesn't overtake the truck till 6pm this means that it takes 8 hours for the van to make up the distance of 80km which means that the van is catching up at a rate of 10km/h. So if the truck is doing 80km/h, THE VAN MUST BE DOING 90km/h. (2nd answer). 5) From 10am till 2pm (the time when the car catches up with the bike) the car will have travelled 480km but during this same period of time the bike will have travelled 160km (40km/h). This means that at 10am the car would have been 320km (480-160) behind the bike. We know from the question that at this time, the car and van are level so the van is also 320km behind the bike. We also know that the vans relative speed to the bike is 50km/h faster so it will take 6.4 hours (6hours 24mins) for the van to catch up with the bike. Which means that THE VAN WILL CATCH UP WITH THE BIKE AT 4:24pm. (3rd Answer). Phew! My (rather tired) brain. |
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