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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.

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

Velocity, v := dr/dt.

Acceleration, a := dv/dt.

Mass flow rate, := dm/dt.

Weighted distance, ŗ := mr.

Momentum, p := dŗ/dt = d(mr)/dt = (mdr + rdm)/dt = m(dr/dt) + r(dm/dt) = mv + rṁ.

If mass is constant, then p = mv.

Force, F  := dp/dt = d(mv + r)/dt = d(mv)/dt + d(r)/dt = (mdv/dt + vdm/dt) + (rd/dt + dr/dt) = ma + v + r + vma + 2v + r, where = d/dt.

If mass is constant, then F = ma.


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

Legerity, u := dw/ds.

Expedience, b := du/ds.

Vass flow rate,  := dn/ds.

Weighted time, ŵ := nw.

Fulmentum, q := dŵ/ds = d(nw)/ds = (ndw + wdn)/ds = n(dw/ds) + w(dn/ds) = nu + wṅ.

If vass is constant, then q = nu.

RushΓ := dq/ds = d(nu + w)/ds = d(nu)/ds + d(w)/ds = (ndu/ds + udn/ds) + (wd/ds + dw/ds) = nb + u + w + u = nb + 2u + w, where  = d/ds.

If vass is constant, then Γ = nb.

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