Recap of Newton's Laws
Laws of Mechanics introduced by Newton assume existing and usage for our purposes so called inertial frames of reference (systems of coordinates) where these laws are held. These frames of reference allow to express the position and velocity of objects under observation.
Also assumed is a concept of force as the action that affects the movements of physical objects.
The important items to be considered when discussing Newton's Laws are the concepts of a material point (an object of zero dimensions but having some inertial mass), space coordinates (usually, Euclidean coordinates in three-dimensional space) and velocities (vectors, components of which are derivatives of corresponding coordinates).
We will usually use symbol m for inertial mass of an object, vector
We will further assume that space coordinates and velocities are functions of time t and
The First Newton's Law states that a material point that is not acted upon by external forces maintains constant velocity, so
dv(t)/dt = 0
This law can be derived from the Second Newton's Law that deals with vectors of forces
This law relates the vector of force and a speed of change in the momentum of an object upon which this force acts.
dp(t)/dt = d(m·v(t))/dt = F(t)
By multiplying all terms by dt, this same law can be also expressed in terms of infinitesimal increment of momentum during an infinitesimal time interval as a result of an impulse of a force, the product of a force vector by that same infinitesimal time interval:
dp(t) = d(m·v(t)) = F(t)·dt
Finally, the Third Newton's Law states that action of one object upon another is always symmetrical. If the force FAB is exerted by object A upon object B, the same by magnitude and opposite by direction force FBA is exerted by object B upon object A.
FAB = −FBA
Important notes
(a) Mass is additive. The mass of two objects combined together is a sum of their masses.
(b) Vectors of forces acting on the same object can be added by the rules of vector algebra, resulting in one vector, whose action is the same as a combination of actions of individual forces.
(c) Each vector equation mentioned above can be broken into three individual equations for each coordinate.
(d) All statements and equations above should be taken as axioms, because they are in good agreement with our day-to-day practice. They represent a theoretical model that we can study further and, based on them, derive numerous properties of moving objects.
Conservation of momentum
Let's do the following experiment.
Two material points A and B are connected with a massless rigid rod. We take this pair and throw it out to open space, where no other forces act on these two material points except a force of one object upon another via a rigid rod that connects them.
Let's analyze the change of momentum of these two objects with time.
Assume, at time t1 the momentum of our objects are pA(t1) and pB(t1). As time goes, our objects move in space in some way that depends on initial push and subsequent interaction with each other via a rod that connects them. At the end of our experiment at time t2 our objects have momentum pA(t2) and pB(t2).
Momentum pA(t2) is the result of an object A initial momentum pA(t1) and combined (that is, integrated) infinitesimal increments of this momentum during the experiment
pA(t2) = pA(t1) + ∫t∈[t1,t2]dpA(t)
As mentioned above, in general,
dp(t) = F(t)·dt
In our case the only force acting on object A is the force FBA(t) exerted by object B.
Therefore,
pA(t2) = pA(t1) + ∫t∈[t1,t2]FBA(t)·dt
Similarly, considering an object B and force acting on it from object A, we have
pB(t2) = pB(t1) + ∫t∈[t1,t2]FAB(t)·dt
Combining these two statements to have a total momentum of the system of these two objects in the beginning and at the end of experiment and taking into consideration the Third Newton's Law
pA(t2)+pB(t2) = pA(t1)+pB(t1)
That is, the total momentum of a closed system (no external forces) is constant.
This is a simple derivation of the Law of Conservation of Momentum.
Granted, it is proven here only in a simple case of two objects, but the proof can be easily extended to a case of any closed system with any number of objects acting upon each other without external forces.
Work and Kinetic Energy
Assume that a material point moves in three-dimensional Euclidean space from time t1 to time t2 along a trajectory described by time-dependent vector r(t) and there is a time-dependent vector of force F(t) acting upon it.
Work performed by this force during the observed time is, by definition,
Wt∈[t1,t2] = ∫t∈[t1,t2]F(t)·dr(t)
Notice that differential of a vector r(t) is a velocity vector v(t) multiplied by differential of time dt.
Also notice that in the formula above we deal with a scalar (dot) product of two vectors - F(t) and dr(t).
According to Newton's Second Law,
dp(t)/dt = F(t)
Also,
dr(t) = v·dt = (p/m)·dt
Therefore,
Wt∈[t1,t2] = ∫t∈[t1,t2](1/m)·p(t)·dp(t)
Integrating this, we obtain
Wt∈[t1,t2]=(1/m)[p²(t2)−p²(t1)]/2=
=m·[v²(t2)−v²(t1)]/2 = T(t2)−T(t1)
where T(t)=m·v²(t)/2 is kinetic energy of an object of mass m moving with velocity v(t).
In other words, work performed by a force equals to an increment of kinetic energy of an object this force acts upon.
Another formulation might be that work performed upon an object is transformed into its kinetic energy.
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