Predicting Change in multiple dimensions
This page discusses the use of momentum to predict change in multi-dimensions and examples of how it is used.
Claimed by rbose7
The Main Idea
The linear momentum, or translational momentum of an object is equal to the product of the mass and velocity of an object. A change in any of these properties is reflected in the momentum.
If the object(s) are in a closed system not affected by external forces the total momentum of the system cannot change.
We can apply these properties to all three dimensions and use momentum to predict the path an object will follow over time by observing the change in momentum in the same way we did in one-dimension.
A Mathematical Model
This change in momentum is shown by the formula:
[math]\displaystyle{ \Delta \overrightarrow{p} }[/math] = [math]\displaystyle{ \overrightarrow{p}_{final}-\overrightarrow{p}_{initial} }[/math] = [math]\displaystyle{ m\overrightarrow{v}_{final}-m\overrightarrow{v}_{initial} }[/math]
Or by relating it to force:
[math]\displaystyle{ \Delta p = F \Delta t\, }[/math]
Relate by Velocity
Given the velocity:
[math]\displaystyle{ \overrightarrow{v} = \left(v_x,v_y,v_z \right) }[/math]
For an object with mass [math]\displaystyle{ \mathbf{m} }[/math]
The object has a momentum of :
[math]\displaystyle{ \overrightarrow{p} }[/math] = [math]\displaystyle{ \overrightarrow{v} * \mathbf{m} }[/math]
= [math]\displaystyle{ \left(v_x,v_y,v_z \right) * \mathbf{m} }[/math]
= [math]\displaystyle{ \left(\mathbf{m} v_x,\mathbf{m} v_y,\mathbf{m} v_z \right) }[/math]
Relate by Force
Given the force:
[math]\displaystyle{ \overrightarrow{F} = \left(F_x,F_y,F_z \right) }[/math]
And change in time:
[math]\displaystyle{ \Delta t }[/math]
[math]\displaystyle{ \Delta p = \overrightarrow{F} \Delta t\, }[/math]
= [math]\displaystyle{ \left(F_x,F_y,F_z \right) * \Delta t }[/math]
= [math]\displaystyle{ \left(\Delta tF_x,\Delta tF_y,\Delta tF_z \right) }[/math]
[math]\displaystyle{ \overrightarrow{p}_{final} = \overrightarrow{p}_{initial} + \Delta p }[/math]
= [math]\displaystyle{ \overrightarrow{p}_{initial} + \left(\Delta tF_x,\Delta tF_y,\Delta tF_z \right) }[/math]
A Computational Model
How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript
A particle with no net force on it:
https://trinket.io/glowscript/b40327b7c7?outputOnly=true
A particle with the force of gravity:
https://trinket.io/embed/glowscript/7065ab6e93?outputOnly=true
A particle launched from a cliff:
Examples
Be sure to show all steps in your solution and include diagrams whenever possible
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