The solution before revision #15539 of , by Julius de Rojas. This is not the current version.

Statement

1.2.16.

Figure 1
Figure 1
The graphs of the coordinate dependence on time is plotted for two particles using two different time scales. The plotted graphs turn out to be the same (Figure 1). One division of the time axis for the graph of the first particle corresponds to , and for the graph of the second particle is . Find the ratio of velocities and the ratio of particle accelerations for point A of the graph.

Solution

We know the instantaneous velocity and acceleration of any object at a given point can be extracted from a plot using


We note that, for any given time, the graphs exactly coincide, so any is exactly the same for both particles, and so . Likewise, for any time interval, we know that , so . We can then say, for an equivalent Point A on their graphs, that:

Using the same logic, we can then say for the accelerations that:

More generally, we can say that, if the time scale divisions are related by , the ratio of velocities are given by:

and the ratio of accelerations is given by:

Answer

The ratio of velocities is

And the ratio of the acceleration is