Momentum at High Speeds: Difference between revisions
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Claimed by Rhiannan Berry Fall 2017 | Claimed by Rhiannan Berry Fall 2017 | ||
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At '''Low velocities''' it is calculated using the formula | At '''Low velocities''' it is calculated using the formula | ||
<math> | |||
<math>KE=\frac{1}{2}mv^2</math> | |||
'''Einstein's Theory of Special Relativity''' | '''Einstein's Theory of Special Relativity''' | ||
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This new adapted formula for '''energy at high speeds''' is: | This new adapted formula for '''energy at high speeds''' is: | ||
<math> | <math>KE=\gamma mc^2</math> | ||
The formula for | The formula for the Lorentz factor, denoted by <math>\gamma</math> is | ||
<math>\gamma = \ | <math>\gamma = \dfrac{1}{1-\beta ^2} \equiv \dfrac{1}{1-\dfrac{v^2}{c^2}} \equiv \dfrac{c}{\sqrt{c^2 - v^2}}</math> | ||
where <math>c</math> | where <math>c</math> | ||
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'''When we put this all together we get''' | '''When we put this all together we get''' | ||
<math> | <math>KE=\left ( \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1\right )mc^2</math> | ||
'''Now for the most important part''' This formula was applied to the momentum formula and we end up with <span style="background-color: #FFFF00">'''the equation for momentum at high speeds'''.</span> | '''Now for the most important part''' This formula was applied to the momentum formula and we end up with <span style="background-color: #FFFF00">'''the equation for momentum at high speeds'''.</span> | ||
<math>\vec{p}=\gamma m \vec{v} = \ | <math>\vec{p}=\gamma m \vec{v} = \dfrac{1}{\sqrt{1-\dfrac{\left | \vec{v} \right | ^2}{c^2}}}m\vec{v}</math> | ||
===A Computational Model=== | ===A Computational Model=== | ||
If you examine the formula for | If you examine the formula for the Lorentz factor, you will see that as the speed of the object approaches the speed of light, gamma becomes exponentially larger and larger. Thus as you approach light speed, a massive amount of Energy is needed and your momentum is huge. A good computer representation of this is: | ||
Model: | Model: Lorentz Factor vs. Speed of Light Graph | ||
[[File:Lambda_vs_Speed_of_Light_Graph.png|200px|thumb alt text]] | [[File:Lambda_vs_Speed_of_Light_Graph.png|200px|thumb alt text]] | ||
Latest revision as of 23:51, 26 November 2017
Claimed by Rhiannan Berry Fall 2017
The Main Idea
Short Description of Topic: In short, momentum varies with speed and as you approach the speed of light, you have to adapt the regular momentum formula to apply to quantum mechanics. This was done by applying Einstein's theory of special relativity to the momentum formula. This gives you the formula for momentum at high speeds.
A Mathematical Model
Momentum at High Speeds is an adaptation of Einstein's formula for Energy at rest:
[math]\displaystyle{ E=mc^2 }[/math]
At Low velocities it is calculated using the formula
[math]\displaystyle{ KE=\frac{1}{2}mv^2 }[/math]
Einstein's Theory of Special Relativity
They found that when you approached the quantum level, the old formula for energy at rest did not apply so it was adapted to quantum mechanics.
This new adapted formula for energy at high speeds is:
[math]\displaystyle{ KE=\gamma mc^2 }[/math]
The formula for the Lorentz factor, denoted by [math]\displaystyle{ \gamma }[/math] is
[math]\displaystyle{ \gamma = \dfrac{1}{1-\beta ^2} \equiv \dfrac{1}{1-\dfrac{v^2}{c^2}} \equiv \dfrac{c}{\sqrt{c^2 - v^2}} }[/math]
where [math]\displaystyle{ c }[/math]
When we put this all together we get
[math]\displaystyle{ KE=\left ( \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1\right )mc^2 }[/math]
Now for the most important part This formula was applied to the momentum formula and we end up with the equation for momentum at high speeds.
[math]\displaystyle{ \vec{p}=\gamma m \vec{v} = \dfrac{1}{\sqrt{1-\dfrac{\left | \vec{v} \right | ^2}{c^2}}}m\vec{v} }[/math]
A Computational Model
If you examine the formula for the Lorentz factor, you will see that as the speed of the object approaches the speed of light, gamma becomes exponentially larger and larger. Thus as you approach light speed, a massive amount of Energy is needed and your momentum is huge. A good computer representation of this is:
Model: Lorentz Factor vs. Speed of Light Graph
Examples
Be sure to show all steps in your solution and include diagrams whenever possible
Simple
Middling
Difficult
References
http://scienceblogs.com/principles/2011/12/03/the-advent-calendar-of-physics-2/