3-Dimensional Position and Motion
Claimed by Benjamin Tasistro-Hart Fall 2016 Motion is defined as a change in position over a time interval. Objects exist in 3-Dimensional space, so it is necessary to know how these objects change position in 3-Dimensional space.
The Main Idea
Objects, exist and move in three dimensions, so it is necessary to understand how it is possible to predict their future position using one of three fundamental principles, in this case the momentum principle. We use an iterative process to calculate and predict the future position of objects. It may seem that using the momentum principle complicates the iterative calculation process, but very rarely do we know only the velocity and acceleration of an object. For this reason, using the momentum principle as the most general form of position update guarantees we can apply it to any situation.
A Mathematical Model
What are the mathematical equations that allow us to model this topic? Position is determined by the net force [math]\displaystyle{ \vec{F}_{net} }[/math] so every type of force, be it a spring, gravity, or electric force, affects the position of an object. It is possible that we are given the net force as number, and we can avoid the iterative calculation of the new magnitude of the spring and gravity force.
Spring Force: [math]\displaystyle{ \vec{F}_{spring} = k_sS\hat{L} }[/math]
Gravity Force: [math]\displaystyle{ \vec{F}_{grav}= }[/math] [math]\displaystyle{ {-G}{m_{1}m_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} }[/math]
Electric Force: [math]\displaystyle{ \vec{F}_{elec}= }[/math] [math]\displaystyle{ {1 \over\ 4\pi\varepsilon_{0}}{q_{1}q_{2} \over\ {\left\vert \vec{r} \right\vert}^2}{\hat{r}} }[/math]
Momentum Principle: [math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math]
The general workflow to solving position-update problems by hand would be as follows:
1. Calculate the current net force [math]\displaystyle{ \vec{F}_{net} }[/math] acting on the system. For this step, remember to update forces which are distant-dependent such as the spring, gravity, and electric forces. 2. Update the new momentum [math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math] 3. Update the new position [math]\displaystyle{ {\vec{r}_{f}}= {\vec{r}_{i}+ \vec{v}_{avg}\Delta t} }[/math]. For this step, [math]\displaystyle{ \vec{v}_{avg} }[/math] can take many forms:
Constant net force: [math]\displaystyle{ \vec{v}_{avg} \approx }[/math] [math]\displaystyle{ \vec{v}_{f}+ \vec{v}_{i} \over\ 2 }[/math]
Non-constant net force: [math]\displaystyle{ \vec{v}_{avg} \approx }[/math] [math]\displaystyle{ \vec{p}_{f}\over\ m }[/math]
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
- How is this topic connected to something that you are interested in?
Everything we do involves three dimensional position and motion. Any force acting upon an object is doing so in three dimensions. If I throw a football, the force I use to throw it is in three dimensions, as is its position and velocity.
- How is it connected to your major?
I am a mechanical engineering major. As mentioned before, whenever an object is acted upon by a force, this happens in three dimensions.
- Is there an interesting industrial application?
Yes! Every force is in three dimensions, as is every object and its movement.