Separating the empirical from the stipulative

What's found and what's (hard-working) convention

Wrong Track: Surely we just decide that forcemass = acceleration is true, just as we decide Δ velocity Δ time = acceleration is true.

Right Lines: Some things are helpful conventions, things we invent to help us describe the world in a special way, so we all agree well enough to test the usefulness of those descriptions. Others are rules about the inventions – discoveries supported by our tests.

Thinking about the learning

Students tend to see all equations as a set of algorithms to be followed to get from a set of inputs to a set of outputs, rather than as relationships between quantities.

Bringing the relationships to life may need you to spend a little more time explaining the nature of the relationships.

Students often don't see any difference in the status of the relationships encoded in the equations, and this can trip them up.

Thinking about the teaching

Everything in kinematics is tautological – that is, true in terms of the definitions. That does not make it useless, empirically, but you should recognise it for what it is – a very sophisticated way of describing the world. This description is highly conventional, but not accidental, and well suited to purpose. In particular, it is an excellent prediction machine, that mimics reality with a high degree of precision, given appropriate inputs to the machine.

But to get started you need to have the accelerations, as well as the initial velocities and displacements. And getting these is a matter of recording the values, and often invoking dynamics.

Dynamics is not a convention: forcemass = acceleration is a statement that could be wrong. It is therefore an empirical statement, and can be, and has been, tested. The fit to the lived-in world is good.

There is also a third kind of relationship, also empirical. These relationships set the forces for this particular situation, so are not universal, like Newton's second law, nor conventional, like kinematics.