Newton’s Second law of Motion – general formulation.
We start with the equation defining Newton’s Second Law of motion in a case where mass is constant
(1)
According to the definition of acceleration we can rewrite this equation into the form
(2)
where
are the initial
and final velocity and 
is the time
during which the change of velocity occurs. The elementary algebra allows us to
write this equation as
(3)
Now we must go forward into Physics and say that the product of mass and velocity of a moving object is defined as the momentum.
(4)
With this definition formula (3) can be written as
(5)
or, dropping out subscripts, we can write equation (5) in a more general form
(6)
If you remember the definition of velocity { link do Displacement and velocity – advanced } http://www.physics-tutorial.net/M1-2-displacement-velocity-advanced.html or acceleration { link to Acceleration } http://www.physics-tutorial.net/M1-3-acceleration.html you realize that equation (6) describes the average force acting during a finite time span. To calculate instantaneous force we must decrease the time span to zero. Such a procedure is simply calculating the derivative of momentum with respect to time
(7)
We can state now the most general formulation of the Newton’s Second law of Motion
The rate of change in momentum is proportional to the net force acting on the object and takes place in the direction of the force
and write it as
(8)
to emphasize that change of momentum is due to the acting force.
Equation (8) describes the situation where the mass of a moving object can change during the motion – the typical conditions for starting rockets.
Further explanation and examples of application of Newton’s Second Law of Motion will be included in the Problems at the end of this Chapter.