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)

 

[astrokreng]

Hello! I would like to provide a response to your question about Newton's second law applied to rocketry.

Firstly, yes, you are correct in your understanding that the equation can be written as F = mdot(v2-v1). This is essentially the same as the last equation given in the quote, F = (m dot * V)e - (m dot * V)0. Both equations are representing the change in momentum, which is equal to the force, and can be written in different ways.

The reason for including mdot twice in the last equation is to account for the fact that the mass flow rate, or mdot, is changing at both the exit of the propulsion device (station "e") and in the free stream (station "0"). This is important to consider because in rocketry, the mass of the propellant (usually a fluid) is constantly changing as it is being ejected from the rocket. So, in order to accurately calculate the change in momentum, we need to account for the change in mass flow rate at both locations.

As for the equation F = d(mv)/dt, this is the same as F = (m dot * V)e - (m dot * V)0. This notation, using "dot" to represent a derivative with respect to time, is commonly used in mathematics and science. It is essentially saying that the force (F) is equal to the change in momentum (d(mv)) over a change in time (dt).

I hope this helps clarify your understanding of Newton's second law applied to rocketry. Let me know if you have any other questions!