Help with Newtons 2nd applied to Rocketry

[JPKelly]

From Newton's second law of motion, we can define a force F to be the change in momentum of an object with a change in time. Momentum is the object's mass m times the velocity V. So, between two times t1 and t2, the force is given by:

F = ((m * V)2 - (m * V)1) / (t2 - t1)

If we keep the mass constant and just change the velocity with time we obtain the simple force equation - force equals mass time acceleration a

F = m * a

If we are dealing with a solid, keeping track of the mass is relatively easy; the molecules of a solid are closely bound to each other and a solid retains its shape. But if we are dealing with a fluid (liquid or gas) and particularly if we are dealing with a moving fluid, keeping track of the mass gets tricky. For a moving fluid, the important parameter is the mass flow rate. Mass flow rate is the amount of mass moving through a given plane over some amount of time. Its dimensions are mass/time (kg/sec, slug/sec, ...) and it is equal to the density r times the velocity V times the area A. Aerodynamicists denote this parameter as m dot (m with a little dot over the top).

m dot = r * V * A

Note: The "dot" notation is used a lot by mathematicians, scientists, and engineers as a symbol for "d/dt", which means the variable changes with a change in time. For example, we can write Newton's second law as either

F = d(mv)/dt or F = (mv)dot

So "m dot" is not simply the mass of the fluid, but is the mass flow rate, the mass per unit time.

Since the mass flow rate already contains the time dependence (mass/time), we can express the change in momentum across the propulsion device as the change in the mass flow rate times the velocity. We will denote the exit of the device as station "e" and the free stream as station "0". Then

F = (m dot * V)e - (m dot * V)0

This is a direct quote from nasa.gov. I understand everything up until the last equation.

Can it not be written like F = mdot(v2-v1) ?

Also, this equation F = d(mv)/dt is the same as this F = (m dot * V)e - (m dot * V)0 ? Or no?

I understand Mass x Velocity is momentum and they want the change in momentum but at the same time mass AND velocity are changing with respect to time.

I guess i don't understand why they wrote it like F = (m dot * V)e - (m dot * V)0 , including mdot twice.

 

[mukundpa]

I think it is more clearer if we write

F = m(dv/dt) + v (dm/dt) = m*(v dot) + v (m dot)