# Formulation of Newton's Laws as foothold principles

#### Prerequisite

In the sections under Physical content of Newton's Laws, we've gone through the conceptual ideas that establish the theoretical framework for our study of motion.  Now let's quantify them and establish principles that can be strong foothold ideas for creating models in specific situations.

In our qualitative discussion of the Physical content of Newton's Laws, we identified the object as what we would focus on, and the influences that caused an object to change its state of motion as interactions with other objects. We then decided that the influences of other objects on an object were shared over the parts of the object. In addition, we decided that when all the influences acting on the object were balanced, it would not change its velocity.

Let's consider two time intervals:  one in which the object is feeling an unbalanced influence, and another when it is feeling only a balance of cancelling influences. We are going to want to quantify everything in sight, so let's give some of these things names.

Let's call an unbalanced influence on an object from other objects acting over a time interval $Δt$ an impulse , $ΔI$. When this impulse is applied to an object of mass $m$, we will share it over the parts of the object by dividing it by the mass.  This produces a change in the velocity of the object. For a simple first case, let's restrict to motion along a single direction, $x$.

$ΔI/m = Δv$.  (During a time interval when an unbalanced influence is acting)

$Δx = \langle v \rangle Δt$.   (During a time interval when no unbalanced influence is acting)

Starting with these basic relations, in the following pages, we go on to quantify what impulse (and the related idea of force) means and extend these to vector equations, creating a framing for describing and building models of motion.

Joe Redish 9/15/11

Article 338