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Newton's Second Law a=F/m assumes that either a body is pointlike or that force is transmitted instantaneously across it. Special relativity which applies to Newton's First Law and which was developed from the assumption of a finite velocity of energy transmission, has enjoyed tremendous success- it is absurd that we have not applied the 'finite velocity of energy transmission' principle to the Second Law as well.

I ended up with F=ma+P/v where P is power and v is velocity of energy transmission which seems too simple and even worse difficulties when extended to energy eqns.

Have I got it wrong in thinking that Newton assumed instantaneous transmission of force across a body?