3-Dimensional Position and Motion - Revision history

Alex: Main Idea rework

2025-12-01T21:26:40Z

Main Idea rework

@@ Line 4: / Line 4: @@
 ==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> \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> \vec{F}_{net} </math> which, by Newton's Second Law of motion, means that the acceleration <math> \vec{a}= </math><math>\vec{F}_{net} \over\ m </math> is constant.
+However, solving problems using vector notation is usually faster and more elegant because vectors combine these three components into a single mathematical object. For example, a velocity vector can be written as:
-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.
+<math> \vec{v} = \frac{\Delta \vec{r}}{\Delta t} = \left\langle \frac{\Delta x}{\Delta t}, \frac{\Delta y}{\Delta t}, \frac{\Delta z}{\Delta t} \right\rangle </math>
-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.
+The foundation of all three-dimensional motion is the position vector,
 ===A Mathematical Model===
-===Kinematic Equations===
+====Kinematic Equations====
-The fundamental equations of motion allow us to observe motion in three dimensions.
+The kinematic equations describe motion of an object when the net force, and therefore the acceleration, is constant. In three dimension, each equations applies to all three components of the motion.
 <math> d= d_0+ v_0t+ </math> <math>at^2 \over\ 2 </math>
@@ Line 23: / Line 34: @@
 <math> v^2= v_0^2 + 2a(d-d_0) </math>
-===Momentum Principle===
+====Momentum Principle====
 ==Examples==

Alex at 20:07, 1 December 2025

2025-12-01T20:07:17Z

← Older revision		Revision as of 16:07, 1 December 2025
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	~~Claimed by Benjamin Tasistro-Hart~~ Fall ~~2016~~		Alexander Haniotis Fall 2025

	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.		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.

Bath3 at 00:45, 28 November 2016

2016-11-28T00:45:11Z

@@ Line 52: / Line 52: @@
 ===Middling===
 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:
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 <math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

Bath3 at 00:39, 28 November 2016

2016-11-28T00:39:50Z

← Older revision		Revision as of 20:39, 27 November 2016
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	<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>		<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

	<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>		<math> {\vec{p}_{avg,2}}= {m{\vec{v}_{f}-\vec{v}_{i} \over\ \Delta t}} </math>

	<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>		<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>
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	===Difficult===		===Difficult===
	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 ~~spring~~ when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,		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> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
	and the moving ball is at location <math> \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> \Delta t </math> of 0.1 s?		and the moving ball is at location <math> \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> \Delta t </math> of 0.1 s?

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	<math> d= d_0+ v_0t </math>		<math> d= d_0+ v_0t </math>

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

	<math> \vec{p} \over\ m </math> <math>= \vec{v} </math>

	<math> d= d_0+ </math> <math> \vec{p}_{0} \over\ m </math> <math> t </math>		<math> d= d_0+ </math> <math> \vec{p}_{0} \over\ m </math> <math> t </math>

Bath3 at 23:02, 27 November 2016

2016-11-27T23:02:57Z

← Older revision		Revision as of 19:02, 27 November 2016
Line 101:		Line 101:

	===Difficult===		===Difficult===
	A cat toy is built from a clump of feathers with mass of 0.014 kg attached to the end of a ~~spring with stiffness 0.915 N/m and relaxed length 0.265 m~~. You hold the toy such that the feathers hang suspended from the spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,		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 spring when your cat swats it around, setting it in motion. At a particular instant the momentum of the ball is <math> \langle −0.02, −0.01, −0.02 \rangle </math> kg · m/s,
	and the moving ball is at location <math> \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> \Delta t </math> of 0.1 s?		and the moving ball is at location <math> \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> \Delta t </math> of 0.1 s?

	~~Remember that since this problem involves the spring force: <math> \vec{v}_{avg} \approx </math> <math>\vec{p}_{f}\over\ m </math>~~

	~~1. Identify a coordinate system. We choose~~ <math> +~~\hat{x}~~ </math> ~~to be to the right <math> +\hat{y} </math> to point up, and <math> +\hat{z} </math> to point out of the page.~~		<math> d= d_0+ v_0t </math>

	~~2. Choose a system and its surroundings. For this example, we choose~~ the ~~system to be the clump~~ of ~~feathers and the surroundings~~ to ~~be the spring and earth.~~		remember the definition of velocity in relation to momentum:

	~~3. Identify external forces which contribute to the net force. In this example, the force of earth and a spring force are the only objects which exert force on the feathers.~~		<math> \vec{p} \over\ m </math> <math>= \vec{v} </math>

	~~Now we can begin to solve this problem. As always, begin from a fundamental principle:~~		<math> d= d_0+ </math> <math> \vec{p}_{0} \over\ m </math> <math> t </math>

	~~<math> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>~~

	~~We know that <math> \vec{F}_{net} </math> is comprised of the spring force <math> \vec{F}_{spring} </math> and the force of earth <math> \vec{F}_{grav} </math>.~~

	~~<math> {\vec{p}_{f}}= {\vec{p}_{i}+ (\vec{F}_{spring}+ \vec{F}_{grav})\Delta t} </math>~~

	~~Recall that <math> \vec{F}_{spring} = k_sS\hat{L} </math>. We need to compute the magnitude of the displacement of the feather clump in order to find S~~
	~~Since the distance between the earth and the feathers is small compared to the radius of earth, we can approximate the magnitude of the force of earth as <math> mg </math>~~


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	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.		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?		#Is there an interesting industrial application?
	Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.		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==		==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.		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.
	== See also ==		== See also ==
	===Further Reading===		===Further Reading===
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	http://farside.ph.utexas.edu/teaching/301/lectures/node33.html		http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
	http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html		http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
			http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Newton.html
	http://kmoddl.library.cornell.edu/what.php		http://kmoddl.library.cornell.edu/what.php

Bath3 at 22:42, 27 November 2016

2016-11-27T22:42:38Z

← Older revision		Revision as of 18:42, 27 November 2016
Line 1:		Line 1:
	Claimed by Benjamin Tasistro-Hart Fall 2016		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.		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==		==The Main Idea==

	Objects, exist and move in three dimensions, and it is possible to reduce the complexities of 3d motion into 3 directions <math> \hat{x}, \hat{y}, \hat{z} </math>. An object in motion has properties along each axis which are independent of other axes.		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> \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> \vec{F}_{net} </math> which, by Newton's Second Law of motion, means that the acceleration <math> \vec{a}= </math><math>\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===		===A Mathematical Model===

			===Kinematic Equations===

	The fundamental equations of motion allow us to observe motion in three dimensions.		The fundamental equations of motion allow us to observe motion in three dimensions.
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	<math> v^2= v_0^2 + 2a(d-d_0) </math>		<math> 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> {\vec{p}_{f}}= {\vec{p}_{i}+ \vec{F}_{net}\Delta t} </math>

	==Examples==		==Examples==
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	(a) What is the average ~~velocity~~ in first interval?		(a) What is the average momentum in first interval?

	(b) the second interval?		(b) the second interval?
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	(a) Begin with the definition of ~~velocity~~:		(a) Begin with the definition of momentum:

			<math> {\vec{p}_{avg,1}}= {m\vec{v}_{avg}} </math>

	<math> {\vec{v}_{avg,1}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>		<math> {\vec{p}_{avg,1}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

	<math> {\vec{v}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>		<math> {\vec{p}_{avg,1}}= \langle 0.0036, -0.0072, -0.0648 \rangle </math>


	(b) Begin with the definition of ~~velocity~~:		(b) Begin with the definition of momentum:

			<math> {\vec{p}_{avg,2}}= {m\vec{v}_{avg}} </math>

	<math> {\vec{v}_{avg,2}}= {{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>		<math> {\vec{p}_{avg,2}}= {m{\vec{r}_{f}-\vec{r}_{i} \over\ \Delta t}} </math>

	<math> {\vec{v}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>		<math> {\vec{p}_{avg,2}}= \langle 0.0072, 0.0171, -0.0648 \rangle </math>


	~~(c) Begin with a fundamental equation of motion~~

	~~<math> {\vec{v}_{2}}= {\vec{v}_{1}+ \vec{a}\Delta t} </math>~~

	~~<math> {\vec{v}_{2}-\vec{v}_{1} \over\ \Delta t}= {\vec{~~a~~}} </math>~~		(c) Begin from a fundamental principle

	~~Remember Newton's second law~~ <math> \vec{F}_{~~net~~}= m\vec{a} </math>		<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{F}_{net}\Delta t} </math>

	<math> m(\vec{v}-\vec{v}~~_0)~~ \over\ t ~~</math> <math>~~ = m(\vec{a}) </math>		<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{F}_{net}} </math>

	~~<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>~~		<math> \langle 0.0018, 0.01215, 0 \rangle = {\vec{F}_{net}} </math>

	~~Substitute the provided values into the symbolic expression and you should arrive at your final answer:~~
	<math> \langle 0.0018, 0.01215, 0 \rangle N= \vec{F}_{net} </math>

	===Difficult===		===Difficult===

Bath3 at 06:22, 27 November 2016

2016-11-27T06:22:49Z

← Older revision		Revision as of 02:22, 27 November 2016
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	(c) Begin with a fundamental equation of motion		(c) Begin with a fundamental equation of motion

	<math> {\vec{v}_{2}}= {\vec{p}_{1}+ \vec{a}\Delta t} </math>		<math> {\vec{v}_{2}}= {\vec{v}_{1}+ \vec{a}\Delta t} </math>

	<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{a}} </math>		<math> {\vec{v}_{2}-\vec{v}_{1} \over\ \Delta t}= {\vec{a}} </math>

	Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>		Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

Bath3 at 06:09, 27 November 2016

2016-11-27T06:09:26Z

← Older revision		Revision as of 02:09, 27 November 2016
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	http://farside.ph.utexas.edu/teaching/301/lectures/node33.html		http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
	http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html		http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
			http://kmoddl.library.cornell.edu/what.php

Bath3 at 06:09, 27 November 2016

2016-11-27T06:09:02Z

← Older revision		Revision as of 02:09, 27 November 2016
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	#Is there an interesting industrial application?		#Is there an interesting industrial application?
	Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which an ability to visualize parts in three dimensions.		Parametric design is related to fabrication, and modern fabrication techniques typically involved creating geometrically complex shapes which 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.
	== See also ==		== See also ==
			===Further Reading===
			http://kmoddl.library.cornell.edu/what.php
			==References==
	http://farside.ph.utexas.edu/teaching/301/lectures/node33.html		http://farside.ph.utexas.edu/teaching/301/lectures/node33.html
	~~===External links===~~		http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
	[http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html]

Bath3 at 05:54, 27 November 2016

2016-11-27T05:54:54Z

← Older revision		Revision as of 01:54, 27 November 2016
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	(c) Begin with a fundamental equation of motion		(c) Begin with a fundamental equation of motion

	<math> {\vec{p}_{2}}= {\vec{p}_{1}+ \vec{~~F}_{net~~}\Delta t} </math>		<math> {\vec{v}_{2}}= {\vec{p}_{1}+ \vec{a}\Delta t} </math>

	<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{~~F}_{net~~}} </math>		<math> {\vec{p}_{2}-\vec{p}_{1} \over\ \Delta t}= {\vec{a}} </math>

	<math> ~~\langle 0.0018, 0.01215, 0 \rangle = {~~\vec{F}_{net}} </math>		Remember Newton's second law <math> \vec{F}_{net}= m\vec{a} </math>

			<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = m(\vec{a}) </math>

			<math> m(\vec{v}-\vec{v}_0) \over\ t </math> <math> = \vec{F}_{net} </math>

			Substitute the provided values into the symbolic expression and you should arrive at your final answer:
			<math> \langle 0.0018, 0.01215, 0 \rangle N= \vec{F}_{net} </math>

	===Difficult===		===Difficult===