Average and instantaneous values

Differences and similarities

Wrong Track: We covered 15 kilometre on our ride in 2 hours so we must have been travelling at 7.5 kilometre / hour.

Right Lines: The average speed is calculated from distanceduration. But it's possible that you weren't travelling at that speed for very much of the journey at all. If you made a movie clip of your cycle computer, showing your current speed, it's likely that for most frames it wouldn't be showing 7.5 kilometre / hour.

Making intelligent choices

Thinking about the learning

Dividing the distance covered during the journey (kilometre) by the duration of the journey (hour) will give an average speed:durationspeed_average = distance

This global value can't be allowed to become confused with the values shown on the measuring instruments carried on the journey, which will show the values at a particular time, not over the whole duration of the journey.

Such right-now representations are true at an instant (so have no necessary connections to the history of the journey up to that instant). Here are some examples: resultant forces, velocities, accelerations, instantaneous speed, and positions.

Thinking about the teaching

You'll need to think about how to refer to measurements that represent an average for a whole journey, or for a significant interval during that journey.

You'll also need to think how to refer to values that represent a right-now value, for a particular clock time, at that instant.

Here are some suggestions, for thoughtful adaptation to your own practice and situation.

Resultant forces, forces, velocities, accelerations and positions are right-now representations, and so are true at an instant (no history). The clock reading is always a series of instants.

So time is always time_{instantaneous}. t stands for this, and only this. Some instant, defined by a clock reading. A unique now-ness. So:

→v_{instantaneous} is a longer way of writing →v.

→s_{instantaneous} is a longer way of writing →s.

→a_{instantaneous} is a longer way of writing →a.

→F_{instantaneous} is a longer way of writing →F.

If we want to deal with them as averages (we'd suggest avoiding this until post-16 studies, if possible) we'd extend the conventions – for example – like this:

→v_average; →s_average; →a_average.