3-Dimensional Position and Motion

In order to be able to calculate the effect of forces on an object, you need to first be able to describe its position and motion in three dimensional space. For locating entities, we have position vectors. The change in position over time creates the velocity vector, which describes motion in space. From here, we can apply three dimensional forces.

The Main Idea

Objects, exist, move and accelerate in three dimensions, so we have to describe them in three dimensions as well.

A Mathematical Model

What are the mathematical equations that allow us to model this topic. For example [math]\displaystyle{ \lt {\frac{d\vec{x}}{dt}},{\frac{d\vec{y}}{dt}},{\frac{d\vec{z}}{dt}}\gt }[/math] is the velocity and [math]\displaystyle{ {\frac{d\vec{(velocity)}}{dt}} }[/math] is the acceleration.

A Computational Model

To program the position in VPython for an object, obj, write obj.pos=(xp,yp,zp). Here xp, yp, and zp are the x, y, and z coordinates, respectively, of the object. Velocity and acceleration are programmed similarly with obj.velocity=(xv,yv,zv) and obj.acceleration=(xa,ya,za). The x, y, and z velocity and acceleration values are xv, yv, and zv and xa, ya, and za respectively.

Examples

Here are a few examples

Simple

obj. is at position (0,0,0) meters, moving at a velocity of (-1, 4, 9) meters per second for n seconds. What is obj.'s position now? (0-n,0+4n,0+9n)=(-n,4n,9n)

Middling

obj. is at position (2,5,8) meters. Acceleration is (2, 9, 0) meters per second squared for 5 seconds. new position= (2,5,8)+(2,9,0)*1/2*5^2= (2,5,8)+(25,112.5,0)=(27,117.5,8)

Difficult

obj. starts at position [math]\displaystyle{ (1,2,1) }[/math] meters with initial velocity [math]\displaystyle{ (1,5,2) }[/math] and an acceleration of [math]\displaystyle{ (-1,4,-2) }[/math]. After four seconds, what is the position? [math]\displaystyle{ position= (initial position) + (initial velocity)*(time) + (acceleration)*1/2(time)^2 }[/math]. [math]\displaystyle{ (1,2,1) + 4*(1,5,2) + 4^2/2*(-1,4,-2)= (1,2,1)+(4,20,8)+(-8,32,-16)=(1+4-8,2+20+32,1+8-16)=(-3,54,-7) }[/math]

Connectedness

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See also

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Further reading

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References

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A Mathematical Model To program the position in VPython for an object, obj, write obj.pos=(xp,yp,zp). Here xp, yp, and zp are the x, y, and z coordinates, respectively, of the object. Velocity and acceleration are programmed similarly with obj.velocity=(xv,yv,zv) and obj.acceleration=(xa,ya,za). The x, y, and z velocity and acceleration values are xv, yv, and zv and xa, ya, and za respectively.