Derivation of the Momentum Principle: Difference between revisions
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==The Main Idea== | ==The Main Idea== | ||
The main idea of the momentum principle is | 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=== | |||
'''Variables:''' | |||
<math> \mathbf{m} = </math> mass in kilograms | |||
<math> \mathbf{r} = </math> distance in meters | |||
<math> \mathbf{t} = </math> time in seconds | |||
<math> \overrightarrow{\mathbf{v}} = \frac{m} {s} </math> | |||
<math> \mathbf{a} = </math> acceleration in <math> \frac{m} {s^2} </math> | |||
<math> \mathbf{g} = 9.8 \frac{m} {s^2}</math> | |||
'''Definition of momentum:''' | |||
Momentum is equal to mass multiplied by velocity. | |||
<math> \overrightarrow{p} = \mathbf{m} * \overrightarrow{\mathbf{v}} </math> | |||
= <math> mass * (x,y,z) </math> | |||
= <math> 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> \Delta{\overrightarrow{p}} = \overrightarrow{\mathbf{F}}_{net} * \Delta{t} </math> | |||
<math> {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> = \overrightarrow{\mathbf{F}}_{grav} = (0,\mathbf{m} * \mathbf{g},0) </math> | |||
<math> \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> \Delta{\overrightarrow{p}} = \overrightarrow{p}_{final} - \overrightarrow{p}_{initial} = \overrightarrow{\mathbf{F}}_{net} * \Delta{t} </math> | |||
<math> \overrightarrow{p}_{final} = \overrightarrow{p}_{initial} + \overrightarrow{\mathbf{F}}_{net} * \Delta{t} </math> | |||
<math> \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> \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> \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> \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> \frac{m} {s^2} </math> | |||
<math> \overrightarrow{\mathbf{v}}_{final} = \overrightarrow{\mathbf{v}}_{initial} + \frac {\Delta{v}} {\Delta{t}} * \Delta{t} </math> | |||
<math> \overrightarrow{\mathbf{v}}_{final} = \overrightarrow{\mathbf{v}}_{initial} + {\Delta{v}} </math> | |||
<math> {\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> {\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> \frac {\Delta{r}} {\Delta{t}} = \overrightarrow{\mathbf{a}} * \Delta{t} </math> | |||
<math> \frac {\mathbf{r}_{final} - \mathbf{r}_{initial}} {\Delta{t}} =\frac {\overrightarrow{\mathbf{v}}_{final} - \overrightarrow{\mathbf{v}}_{initial}} {\Delta{t}} * \Delta{t} </math> | |||
<math> \mathbf{r}_{final} - \mathbf{r}_{initial} =\overrightarrow{\mathbf{v}}_{final} - \overrightarrow{\mathbf{v}}_{initial} * \Delta{t} </math> | |||
<math> \mathbf{r}_{final} - \mathbf{r}_{initial} =\Delta{v} * \Delta{t} </math> | |||
<math> \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. | |||
==History== | |||
Put this idea in historical context. Give the reader the Who, What, When, Where, and Why. | |||
== See also == | |||
Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context? | |||
===Further reading=== | |||
Books, Articles or other print media on this topic | |||
===External links=== | |||
[http://www.scientificamerican.com/article/bring-science-home-reaction-time/] | |||
==References== | |||
This section contains the the references you used while writing this page | |||
[[Category:Momentum]] | |||
==Examples== | ==Examples== | ||
Revision as of 17:52, 4 December 2015
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
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.
History
Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
See also
Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?
Further reading
Books, Articles or other print media on this topic
External links
References
This section contains the the references you used while writing this page
Examples
Be sure to show all steps in your solution and include diagrams whenever possible
Simple
Middling
Difficult
Connectedness
- How is this topic connected to something that you are interested in?
- How is it connected to your major?
- Is there an interesting industrial application?
History
Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
See also
Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?
Further reading
Books, Articles or other print media on this topic
External links
References
This section contains the the references you used while writing this page