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

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

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