Meeting point for the way-time function: different starting points

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"Tim gets on a train in Cologne at 6:00 p.m. and drives to Hamburg at 160 km / h. At the same time, Jonas gets on a train in Hamburg and comes towards Tim at 120 km / h. When do the two meet? "That kind of task where the point is based on information about the different ones Starting points, the path, the time and the speed of calculating the point of contact between two objects are in puzzle and math books very popular. But how can function equations be set up here and how is the task solved?

If trains approach each other, they will meet at some point.
If trains approach each other, they will meet at some point.

Create functions for the different starting points, routes and times

If you want to solve a task of the type described, it is essential that you have information about the path to be covered between the have different starting points, as well as the speed of the objects, i.e. the time in which they have a certain path return.

  • If you take the example from the introduction, you have to know that Cologne and Hamburg are about 450 kilometers apart. Now create a function term that describes Tim's movement: f (x) = x * 160km / h. x is the time that Tim is on the train, f(x) is therefore the distance traveled. For Jonas the term is f(x) = x * 120km / h.
  • The two functional terms do not yet take into account that the boys move in different directions. To take this into account, Jonas' movement is subtracted from the distance to Tim, i.e. f(x) = 450km - x * 120 km / h.

How to calculate the meeting point

  • To determine the meeting point, you have to equate both function terms. x * 160km / h = 450km - x * 120km / h. Now calculate + x * 120km / h on both sides. Now the equation is: x * 280km / h = 450km. Now divide by 280km. What remains is: x * h = 1.61. After 1.6 hours, as 96 minutes, the two trains meet.
  • You can the Functions Incidentally, also transferred to a coordinate system. If they do so, you will find that the two are different Straight lines cut in one point. This is the meeting point of the two trains. The different starting points are taken into account by shifting Jonas on the Y-axis (distance) by 450km. The X-axis describes the time. The speed is the respective slope of the function.
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