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Derivation of Newton’s second law

It is often said that Newton’s laws are laws of nature, which can only be determined by observation. That’s true in the sense that the definitions required are based on inductive reasoning. However, once these definitions are in hand, it should be a deductive science. This is classical science, in the sense of Euclid, Archimedes, Plato, and Aristotle.

Here is a derivation of Newton’s second law in space-time, with distance, s, time, t, and mass, m:

Velocity, v := ds/dt.

Acceleration, a := dv/dt.

Mass flow rate, := dm/dt.

Weighted distance, ș := ms.

Momentum, p := dș/dt = d(ms)/dt = (mds + sdm)/dt = m(ds/dt) + s(dm/dt) = mv + sṁ.

If mass is constant, then p = mv.

Force, F  := dp/dt = d(mv + sṁ)/dt = d(mv)/dt + d(sṁ)/dt = (mdv/dt + vdm/dt) + (sd/dt + ds/dt) = ma + vṁ + sṃ + ṁvma + 2ṁv + sṃ, where = d/dt.

If mass is constant, then F = ma.

Here is a derivation of Newton’s second law in time-space, with vass, n = 1/m:

Allegrity, u := dt/ds.

Modulation, b := du/ds.

Vass flow rate,  := dn/ds.

Weighted durationț := nt.

Celentum, q := dț/ds = d(nt)/ds = (ndt + tdn)/ds = n(dt/ds) + t(dn/ds) = nu + tṅ.

If vass is constant, then q = nu.

SurgeΓ := dq/ds = d(nu + tṅ)/ds = d(nu)/ds + d(tṅ)/ds = (ndu/ds + udn/ds) + (td/ds + dt/ds) = nb + uṅ + tṇ + ṅu = nb + 2ṅu + tṇ, where  = d/ds.

If vass is constant, then Γ = nb.

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