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, and there are several conceptual models in physics such as kinematics or the momentum principle which describe motion. Because kinematics and the momentum principle are both vector quantities, it is possible to reduce the complexities of 3d motion into 3 directions [math]\displaystyle{ \hat{x}, \hat{y}, \hat{z} }[/math]. The kinematic equations are most useful when the object under observation is subject to a constant force [math]\displaystyle{ \vec{F}_{net} }[/math] which, by Newton's Second Law of motion, means that the acceleration [math]\displaystyle{ \vec{a}= }[/math][math]\displaystyle{ \vec{F}_{net} \over\ m }[/math] is constant.

The use of the momentum principle is most applicable because we can apply it to any situation. 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 both models, an object in motion has properties along each axis which are independent of other axes allowing us to decompose three-dimensional motion into three one-dimensional problems.

A Mathematical Model

Kinematic Equations

The fundamental equations of motion allow us to observe motion in three dimensions.

[math]\displaystyle{ d= d_0+ v_0t+ }[/math] [math]\displaystyle{ at^2 \over\ 2 }[/math]

[math]\displaystyle{ v= v_0+at }[/math]

[math]\displaystyle{ v^2= v_0^2 + 2a(d-d_0) }[/math]

Momentum Principle

The fundamental equations of motion allow us to observe motion in three dimensions.

[math]\displaystyle{ {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} }[/math]

Examples

Here are a few examples:

Simple

At t = 10.0 seconds a mass of 3 kg has velocity of [math]\displaystyle{ \langle 12, 27, −8 \rangle }[/math] m/s. At t = 10.1 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?

[math]\displaystyle{ \vec{v}= \vec{v}_0+\vec{a}t }[/math]

[math]\displaystyle{ \vec{v}-\vec{v}_0 \over\ t }[/math] [math]\displaystyle{ = \vec{a} }[/math]

Remember Newton's second law [math]\displaystyle{ \vec{F}_{net}= m\vec{a} }[/math]

[math]\displaystyle{ m(\vec{v}-\vec{v}_0) \over\ t }[/math] [math]\displaystyle{ = m(\vec{a}) }[/math]

[math]\displaystyle{ m(\vec{v}-\vec{v}_0) \over\ t }[/math] [math]\displaystyle{ = \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

A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a string. You hold the toy such that the feathers hang suspended from the string when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is [math]\displaystyle{ \langle −0.02, −0.01, −0.02 \rangle }[/math] kg · m/s, and the moving ball is at location [math]\displaystyle{ \langle −0.2, −0.61, 0 \rangle }[/math] m relative to an origin located at the base of the spring. What is the position of the ball 0.1 s later with a time-step [math]\displaystyle{ \Delta t }[/math] of 0.1 s?

[math]\displaystyle{ d= d_0+ v_0t }[/math]

remember the definition of velocity in relation to momentum: [math]\displaystyle{ \vec{p} \over\ m }[/math] [math]\displaystyle{ = \vec{v} }[/math]

[math]\displaystyle{ d= d_0+ }[/math] [math]\displaystyle{ \vec{p}_{0} \over\ m }[/math] [math]\displaystyle{ t }[/math]

Difficult

An acorn of mass 2.7 g is acted upon by the Earth, air resistance, and a strong wind. Below are recorded times and positions:

First Interval:

At t = 1.56 s, the position was [math]\displaystyle{ \langle 4.22, 2.45, −9.63 \rangle }[/math] m.

At t = 1.59 s, the position was [math]\displaystyle{ \langle 4.26, 2.37, −10.35 \rangle }[/math] m.

Second Interval:

At t = 3.56 s, the position was [math]\displaystyle{ \langle 8.09, 6.18, -58.35 \rangle }[/math] m.

At t = 3.59 s, the position was [math]\displaystyle{ \langle 8.17, 6.37, -59.07 \rangle }[/math] m.

(a) What is the average momentum in first interval?

(b) the second interval?

(c) What was the average force applied during these two intervals?

(a) Begin with the definition of momentum:

[math]\displaystyle{ {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle }[/math]

(b) Begin with the definition of momentum:

[math]\displaystyle{ {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,2}}= {m{\vec{v}_{f}-\vec{v}_{i} \over\ \Delta t}} }[/math]

[math]\displaystyle{ {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle }[/math]

(c) Begin from a fundamental principle

[math]\displaystyle{ {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} }[/math]

[math]\displaystyle{ {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} }[/math]

[math]\displaystyle{ \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} }[/math]

Connectedness

How is this topic connected to something that you are interested in?

Everything we do involves three dimensional position and motion, so it is useful to understand how matter moves in three dimensions. I'm interested in parametric design. so understanding how things change in three dimensions is critical for good design.

How is it connected to your major?

I study architecture, so an awareness of three dimensional space is critical to designing well. As mentioned before, I'm interested in parametric design which when used to create complex shapes (like those buildings which have a flowy diagrid [see Norman Foster's British Museum]) demands an understanding of the way matter moves through space.

Is there an interesting industrial application?

Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which demand an ability to visualize parts in three dimensions.

History

Compared to Newtonian physics, the field of kinematics is a relatively recent exploration. André-Marie Ampère wrote in his 1834 essay Essai sur la philosophie des sciences about the need for a field of science which analyzes motion independent of the forces. Work continued throughout the nineteenth century under Franz Reuleaux who is considered the father of modern kinematics. In 1666, Newton formulated early versions of his three laws of motion, of which the firstl aw describes the momentum principle. Two decades later, he would publish Principia which is often cited as one of greatest scientific books ever written.

3-Dimensional Position and Motion

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