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
The following is code for a simple computational model showing the effects of a constant force on a mass:
https://trinket.io/embed/glowscript/75acfdd1c6
Examples
Here are a few examples:
Simple
At t = 17.0 seconds an object with mass 3 kg was observed to have a velocity of [math]\displaystyle{ \langle 12, 27, −8 \rangle }[/math] m/s. At t = 17.1 seconds its velocity was [math]\displaystyle{ \langle 24, 19, 22 \rangle }[/math] m/s. What was the average (vector) net force acting on the object?
Begin from a fundamental principle:
[math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math]
[math]\displaystyle{ {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} }[/math]
[math]\displaystyle{ {\vec{p}_{f}-\vec{p}_{i} \over\ \Delta t}= \vec{F}_{net} }[/math]
Substitute the provided values into the symbolic expression and you should arrive at your final answer:
[math]\displaystyle{ \left\langle 360,\ -240,\ 900\right\rangle N = \vec{F}_{net} }[/math]
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
A satellite of mass [math]\displaystyle{ 2.5\times10^4 }[/math] kg coasts with a constant velocity of [math]\displaystyle{ \langle 0, 23, 0 \rangle }[/math] km/s. As it passes the location [math]\displaystyle{ \langle 6, 8, 0 \rangle }[/math] km a force of [math]\displaystyle{ \langle 8\times10^5, 0, 0 \rangle }[/math] N is applied by the booster rockets for 23.5 s. The ejected gases have a mass that is small compared to the mass of the spacecraft. After 23.5, the satellite resumes coasting. Where are you an hour later? (Think about what approximations or simplifying assumptions you made in your analysis. Also think about the choice of system: what are the surroundings that exert external forces on your system?)
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.