Derivation of the Momentum Principle
By kpatel358
claimed by emccoy35 (fall 16)
The Momentum Principle has many different equations. The easiest way to learn these is not memorize each of them, but rather understand all the derivations involved with it. All of physics connect, but learning how they connect is more important than learning all the different equations.
The Main Idea
Knowing the Momentum Principle and the derivation of the Momentum principle allows you to better understand the connections between variables and how they relate to each other. This provides a stronger understanding of momentum and other physics principles.
A Mathematical Model
Variables:
[math]\displaystyle{ \mathbf{m} = }[/math] mass in kilograms
[math]\displaystyle{ \mathbf{r} = }[/math] distance in meters
[math]\displaystyle{ \mathbf{t} = }[/math] time in seconds
[math]\displaystyle{ \overrightarrow{\mathbf{v}} = \frac{m} {s} }[/math]
[math]\displaystyle{ \mathbf{a} = }[/math] acceleration in [math]\displaystyle{ \frac{m} {s^2} }[/math]
[math]\displaystyle{ \mathbf{g} = 9.8 \frac{m} {s^2} }[/math]
Definition of momentum:
Momentum is equal to mass multiplied by velocity.
[math]\displaystyle{ \overrightarrow{p} = \mathbf{m} * \overrightarrow{\mathbf{v}} }[/math] = [math]\displaystyle{ mass * (x,y,z) }[/math] = [math]\displaystyle{ kg * \frac{m} {s} }[/math]
Momentum Principle:
The momentum principle states that the change in momentum of a system is equal to the net force on that system multiplied by the change in time. Another term for the net force multiplied by the change in time is impulse, which goes back to saying that the change in momentum is equal to the impulse.
[math]\displaystyle{ \Delta{\overrightarrow{p}} = \overrightarrow{\mathbf{F}}_{net} * \Delta{t} }[/math]
[math]\displaystyle{ {Impulse} = \overrightarrow{\mathbf{F}}_{net} * \Delta{t} }[/math]
The net force is found my determining all of the forces acting on the system. The general force to take into account is the gravitational force. There is only a Y component for the gravitational force, but force can have all three components.
Gravitational Force [math]\displaystyle{ = \overrightarrow{\mathbf{F}}_{grav} = (0,\mathbf{m} * \mathbf{g},0) }[/math]
[math]\displaystyle{ \mathbf{\overrightarrow{F}}_{net} = (0,9.8,0) N }[/math]
For the purpose of derivation, the best way to recognize the relationship between different equations is to use the variables instead of numbers and find all of the different ways the momentum principle can be manipulated.
[math]\displaystyle{ \Delta{\overrightarrow{p}} = \overrightarrow{p}_{final} - \overrightarrow{p}_{initial} = \overrightarrow{\mathbf{F}}_{net} * \Delta{t} }[/math]
[math]\displaystyle{ \overrightarrow{p}_{final} = \overrightarrow{p}_{initial} + \overrightarrow{\mathbf{F}}_{net} * \Delta{t} }[/math]
[math]\displaystyle{ \mathbf{m} * \overrightarrow{\mathbf{v}}_{final} = \mathbf{m} * \overrightarrow{\mathbf{v}}_{initial} + \overrightarrow{\mathbf{F}}_{net} * \Delta{t} }[/math]
After manipulating the equation to the one above, it's possible to divide both sides by mass to get the equation:
[math]\displaystyle{ \overrightarrow{\mathbf{v}}_{final} = \overrightarrow{\mathbf{v}}_{initial} + \frac {\overrightarrow{\mathbf{F}}_{net}} {\mathbf{m}} * \Delta{t} }[/math]
Another basic equation in Physics is that force is equal to mass multiplied by acceleration.
[math]\displaystyle{ \mathbf{\overrightarrow{F}} = \mathbf{m} * \overrightarrow{\mathbf{a}} }[/math]
So from this equation, one can see that force divided by mass is just acceleration.
[math]\displaystyle{ \overrightarrow{\mathbf{v}}_{final} = \overrightarrow{\mathbf{v}}_{initial} + \overrightarrow{\mathbf{a}} * \Delta{t} }[/math]
Acceleration is the change in velocity over the change in time in [math]\displaystyle{ \frac{m} {s^2} }[/math]
[math]\displaystyle{ \overrightarrow{\mathbf{v}}_{final} = \overrightarrow{\mathbf{v}}_{initial} + \frac {\Delta{v}} {\Delta{t}} * \Delta{t} }[/math]
[math]\displaystyle{ \overrightarrow{\mathbf{v}}_{final} = \overrightarrow{\mathbf{v}}_{initial} + {\Delta{v}} }[/math]
[math]\displaystyle{ {\Delta{v}} = \overrightarrow{\mathbf{v}}_{final} - \overrightarrow{\mathbf{v}}_{initial} }[/math]
Going back to the equation with acceleration, one can see that the change in velocity is equal to acceleration multiplied by the change in time.
[math]\displaystyle{ {\Delta{v}} = \overrightarrow{\mathbf{a}} * \Delta{t} }[/math]
Because velocity is the change in distance over the change in time. we can see that
[math]\displaystyle{ \frac {\Delta{r}} {\Delta{t}} = \overrightarrow{\mathbf{a}} * \Delta{t} }[/math]
[math]\displaystyle{ \frac {\mathbf{r}_{final} - \mathbf{r}_{initial}} {\Delta{t}} =\frac {\overrightarrow{\mathbf{v}}_{final} - \overrightarrow{\mathbf{v}}_{initial}} {\Delta{t}} * \Delta{t} }[/math]
[math]\displaystyle{ \mathbf{r}_{final} - \mathbf{r}_{initial} =\overrightarrow{\mathbf{v}}_{final} - \overrightarrow{\mathbf{v}}_{initial} * \Delta{t} }[/math]
[math]\displaystyle{ \mathbf{r}_{final} - \mathbf{r}_{initial} =\Delta{v} * \Delta{t} }[/math]
[math]\displaystyle{ \mathbf{r}_{final} = \mathbf{r}_{initial} + \Delta{v} * \Delta{t} }[/math]
History
The idea of momentum has been talked about by different philosophers and scientists for many years before there was a mathematical representation of the principle. One of the first to state the definition of momentum in the way that is correct and used to day was John Wallis. He had no mathematical representation or derivation of the Momentum Principle but he had the concepts and ideas down. The first known mathematical representation is tough to say for certain but a mathematical representation and derivation of the Momentum Principle was starting to be seen in text like Jenning's 'Miscellanea' and other works of the time period. That time period is around 1720. Around then, mathematical representations of the variables in the Momentum Principle were starting to show up in text and was a few years later published in Newton's 'Principia Mathematica.'
Connectedness
With this derivation, many of the updating formulas do not seem as intimidating. It is a lot easier to understand physics after knowing how everything is connected, and from this many can see that some of the main parts of physics is velocity, time, distance, force, and momentum. The Momentum Principle will be important in any major involving physics. For example, an engineer designing a car or a plane must have extensive knowledge of the momentum principle in order to know exactly how the vehicle and materials it is made of will react under certain conditions. An understanding of the Derivation of the Momentum Principle is necessary for this to be possible.
References
Chabay, Ruth W., and Bruce A. Sherwood. Matter & Interactions. 4th ed. Hoboken, NJ: John Wiley & Sons, 2015. Print.
"Elastischer stoß" by Simon Steinmann - Own work. Licensed under CC BY-SA 2.5 via Commons - https://commons.wikimedia.org/wiki/File:Elastischer_sto%C3%9F.gif#/media/File:Elastischer_sto%C3%9F.gif
"Newtons cradle animation book 2" by DemonDeLuxe (Dominique Toussaint) - Image:Newtons cradle animation book.gif. Licensed under CC BY-SA 3.0 via Commons - https://commons.wikimedia.org/wiki/File:Newtons_cradle_animation_book_2.gif#/media/File:Newtons_cradle_animation_book_2.gif
Dugas, René (1988). A history of mechanics. Translated into English by J.R. Maddox (Dover ed.). New York: Dover Publications.
Jennings, John (1721). Miscellanea in Usum Juventutis Academicae. Northampton: R. Aikes & G. Dicey.