iSoul In the beginning is reality

Travel time and temporal displacement

There’s a basic distinction between the distance traveled and the displacement. There should be a corresponding distinction between the travel time and the temporal displacement – which I’ll call the distimement (dis-time-ment vs. dis-place-ment).

The travel time is the total duration of the trip, and the travel distance is the total distance traveled (odometer distance). The displacement is the result of the change in position, the resultant position vector. Similarly, the distimement is the result of the change of the point in time, the resultant time position vector.

A round-trip begins and ends at the same place; its displacement is zero. However, the place is not exactly the same because changes are always occurring – you can’t step in the same river twice (Heraclitus). Temporally, a round-trip begins at a certain point in time – an event – and returns to the ‘same’ event so that its time displacement is zero. It’s not exactly the same event but a parallel event such as opening the door to the same home.

Sometimes in my career when I went on official travel, I would take a personal side trip at my own expense. In order to make the official travel voucher as straightforward as possible, I would begin and end my side trip at the same time of day on a subsequent day. For example, if my official business was complete at 5:00 pm Thursday, instead of taking a flight home I would switch to personal time and go off a few days on my own. Then at 5:00 pm on, say, Saturday, I would switch back to official time and fly home on official travel. The time displacement of the side trip was zero, which made it easy to remove the personal time from the official per diem allowance.

Speed is the ratio of the distance traveled to the travel time, that is, distance over duration. Velocity is the ratio of the change in position – the displacement – to the travel time. What I’ve called celerity should be defined analogously as the ratio of the change in temporal position – the distimement – to the distance traveled.

The dependent variable in velocity is the displacement. Similarly, the dependent variable in celerity is the distimement. The independent variable in velocity is the travel time for a single motion but for multiple motions the independent variable is the magnitude of the distimement, not the total travel time.

Similarly, the independent variable in celerity is the travel distance for a single motion but for multiple motions the independent variable is the magnitude of the displacement, not the total distance traveled. Thus the resultant of several velocities or celerities is their vector sum.

If someone travels 30 km North in 4 hr, then 40 km East in 3 hr, their displacement is 50 km Northeast at 53 degrees from the North and their distimement is 5 hr Northeast in a course 37 degrees clockwise from the North. Their velocities are 30 km/4 hr = 7.5 km/hr North and 40 km/3 hr = 13.3 km/hr East.

The trip took 3 + 4 = 7 hr. So is their resultant velocity 50 km/7 hr? Or is the resultant velocity the displacement divided by the magnitude of the distimement, 50 km/5 hr = 10 km/hr? It is the latter. So the distimement is really not new.

Similarly, their celerities are 4/30 = 8 min/km North and 3/40 = 4.5 min/km East. The distance traveled is 30 + 40 = 70 km. Is their resultant celerity 5 hr/70 km? No, it’s the distimement divided by the magnitude of the displacement, 5 hr/50km = 6 min/km.

One might define the effective velocity of a trip as the displacement divided by the total travel time. And the effective celerity of a trip might be defined as the distimement divided by the total travel distance. But these are different from the resultant vectors of velocity and celerity.

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