Why is time a variable

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The general case - variable acceleration

The acceleration is not always constant during a process. For example, a trip on the subway can be roughly divided into three intervals:

  • in the acceleration process
  • driving at constant speed
  • in the braking process at the next stop

If one assumes that the acceleration is constant in each of these three sub-areas of the movement, then these can be treated individually according to one of the special cases listed above. The necessary initial conditions result from the final speed or the location of the previous section.

Task: subway ride

A subway arrives at the stop and accelerates up to the speed of. Then it drives on at a constant speed. When do you have to start slowing it down if you want it to come to a stop at the next station that is away? It can be braked with.


1st interval: Accelerate from up to
The train accelerates with it. It accelerates until it has reached the driving speed of. Therefore, with the start time and a start speed, the following applies: This results in the point in time: Until the subway has accelerated to its travel speed, it covers the following distance with the start path:
3. Interval: Decelerate from to
Which way is required to bring the subway to a standstill from when there is a deceleration? : During this time, the train continues the route: The speed of the subway at the point in time is, therefore.
2. Interval: Driving at constant speed from to
Since the two underground stations are distant and are covered when accelerating or braking, the train still has to travel the way in the second time interval: It needs the time to do this: The braking process must therefore be started at the following point in time:

General case

Normally, however, the subway will not be able to travel at constant speeds on its way because it has to travel more slowly in some places because of a curve or a construction site. The acceleration during start-up and braking is also not constant.

Work order: Work order 1

Start the project with the blue arrow key. The acceleration can be adjusted with the slider. The graphs are drawn by pressing the start button.

So the acceleration is a function of time. In order to answer the question about the speed and the location, the acceleration has to be integrated. This does not always have to be fundamentally possible for accelerations that can be changed at will. In this case we come back to the previously considered decomposition of the movement into sub-intervals. If these are chosen to be sufficiently small, the movement can be calculated approximately. This method is called numerical calculation.

Work order: Work order 2

Solve the subway task using the acceleration-time and speed-time graphs by using the areas under the graphs to calculate the speed and distance!

Solution to work order 2