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Newton originally wrote his second law as:

"The rate of change of momentum of a body is directly proportional to the resultant force applied to the body, and is in the same direction as the force."

This definition implies,

$F=k\frac{dp}{dt}$ where $k$ is the constant of proportionality, and $p$ is the object's momentum.

From this relationship, why is it that we are able to deduce that $k=1$, and consequently form the equation:

$F=ma$. (mass being constant).

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    $\begingroup$ That is true in SI system of units, if you change the system, the proportionality will change $\endgroup$ – Oswald Jan 7 '16 at 4:55
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    $\begingroup$ @TheGhostOfPerdition I don't think that's true. An equation such as $F = m a$ is not written in any unit system. Setting $k=1$ is a definition of either $F$ or $p$, or if force and momentum are already defined then $k=1$ is an experimental fact. $\endgroup$ – DanielSank Jan 7 '16 at 5:04
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    $\begingroup$ @DanielSank , If we use grams instead of kgs then the proportionality would be equivalent to (1/1000), I think the system of units were chosen so as to call '$1$ kg or mass accelerated at $1m/s^2$' as 1 newton, to get rid of the proportionality constant $\endgroup$ – Oswald Jan 7 '16 at 5:15
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    $\begingroup$ One should think of $F=\dfrac{dp}{dt}$ as the definition of force just as $v=\dfrac{dx}{dt}$ is the definiton of velocity. The constant of proportionality then by definition is 1. $\endgroup$ – Omar Nagib Jan 7 '16 at 5:44
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    $\begingroup$ @CuriousOne no, I don't think so. It seems like a perfectly fine science question to me. It may be a duplicate of this, though - if not a duplicate, at least closely related. $\endgroup$ – David Z Jan 7 '16 at 8:41
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At the time of Isaac Newton there was no defined unit for force. This gave him the freedom of choosing any unit which was convenient.So he defined a unit of force such that the proportionality constant $k$ becomes 1 which simplified the formula.

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  • $\begingroup$ Agree. I'd even say that at Isaac Newton time there was no force unit named 'Newton' :). He just didn't know that 1 kg m / s2 equals exactly 1 N. (joke, but maybe not joke). $\endgroup$ – dmafa Jun 3 '16 at 18:43

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