Derivation of the Momentum Principle

By kpatel358

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

The main idea of deriving different ideas from the momentum principle is to see the relationships in different views. After understanding the different connections, it is easier to recognize how the variables connect to each other.

A Mathematical Model

Newtons cradle animation book 2

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]

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.

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