Derivation of Average Velocity - Revision history

YorickAndeweg at 21:35, 8 August 2019

2019-08-08T21:35:16Z

← Older revision		Revision as of 17:35, 8 August 2019
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	This ~~is a short page presenting the mathematical~~ derivation of the formula ~~<math>\vec{v}_{avg} = \frac{v_i + v_f}{2}</math>~~.		Claimed by Gahan Wang (Fall 2016)
			:The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.

	==The Main Idea==		==The Main Idea==

YorickAndeweg at 21:18, 8 August 2019

2019-08-08T21:18:05Z

← Older revision		Revision as of 17:18, 8 August 2019
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	~~Claimed by Gahan Wang (Fall 2016)~~		This is a short page presenting the mathematical derivation of the formula <math>\vec{v}_{avg} = \frac{v_i + v_f}{2}</math>.
	~~:The basic formula for average velocity~~ is ~~easy to understand, but how exactly can you prove this formula? This reference aims to show how using~~ derivation ~~through geometric, algebraic, and computer models proves~~ the formula ~~for average velocity~~.

	==The Main Idea==		==The Main Idea==

Gwang307 at 04:58, 28 November 2016

2016-11-28T04:58:18Z

← Older revision		Revision as of 00:58, 28 November 2016
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	Claimed by Gahan Wang (Fall 2016)		Claimed by Gahan Wang (Fall 2016)
	=The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.		:The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.

	==The Main Idea==		==The Main Idea==

Gwang307 at 04:58, 28 November 2016

2016-11-28T04:58:11Z

← Older revision		Revision as of 00:58, 28 November 2016
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	Claimed by Gahan Wang (Fall 2016)		Claimed by Gahan Wang (Fall 2016)
	The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.		=The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.

	==The Main Idea==		==The Main Idea==

Gwang307 at 04:58, 28 November 2016

2016-11-28T04:58:03Z

← Older revision		Revision as of 00:58, 28 November 2016
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	:Claimed by Gahan Wang (Fall 2016)		Claimed by Gahan Wang (Fall 2016)
	:The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.		The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.

	==The Main Idea==		==The Main Idea==

Gwang307 at 04:57, 28 November 2016

2016-11-28T04:57:52Z

← Older revision		Revision as of 00:57, 28 November 2016
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	Claimed by Gahan Wang (Fall 2016)		:Claimed by Gahan Wang (Fall 2016)
	:The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.		:The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.

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	===Further Reading===		===Further Reading===
	https://www.youtube.com/watch?v=T_RjQAInWBc		:https://www.youtube.com/watch?v=T_RjQAInWBc
	https://www.boundless.com/physics/textbooks/boundless-physics-textbook/kinematics-2/speed-and-velocity-36/average-velocity-a-graphical-interpretation-210-6244/		:https://www.boundless.com/physics/textbooks/boundless-physics-textbook/kinematics-2/speed-and-velocity-36/average-velocity-a-graphical-interpretation-210-6244/

Gwang307 at 04:57, 28 November 2016

2016-11-28T04:57:19Z

← Older revision		Revision as of 00:57, 28 November 2016
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	==Connectedness==		==Connectedness==
	:Using basic, fundamental mathematical variables to prove physics equations shows the connection between math and science and how the same concept of limits and derivatives applies to an important, primary scientific principle in average velocity.		:Using basic, fundamental mathematical variables to prove physics equations shows the connection between math and science and how the same concept of limits and derivatives applies to an important, primary scientific principle in average velocity. One could also use Computer Science to create a program that could derive the average velocity formula.

			===Further Reading===
			https://www.youtube.com/watch?v=T_RjQAInWBc
			https://www.boundless.com/physics/textbooks/boundless-physics-textbook/kinematics-2/speed-and-velocity-36/average-velocity-a-graphical-interpretation-210-6244/


	===External links===		===External links===
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	:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity		:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
	:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html		:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html
			http://www.luc.edu/faculty/dslavsk/courses/ntsc395/classnotes/ntsc395equations.pdf

	==References==		==References==

Gwang307 at 03:23, 28 November 2016

2016-11-28T03:23:11Z

← Older revision		Revision as of 23:23, 27 November 2016
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	Claimed by Gahan Wang (Fall 2016)		Claimed by Gahan Wang (Fall 2016)
	The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.		:The basic formula for average velocity is easy to understand, but how exactly can you prove this formula? This reference aims to show how using derivation through geometric, algebraic, and computer models proves the formula for average velocity.

	==The Main Idea==		==The Main Idea==

Gwang307 at 03:22, 28 November 2016

2016-11-28T03:22:41Z

← Older revision		Revision as of 23:22, 27 November 2016
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	:<math>= \frac{v_f+v_i}{2} </math>		:<math>= \frac{v_f+v_i}{2} </math>
	:Solving the integral results in the original equation, <math> \frac{v_f + v_i}{2} </math>		:Solving the integral results in the original equation, <math> \frac{v_f + v_i}{2} </math>

	~~===A Computational Model===~~


	==Examples==		==Examples==
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	:Description of Motion. N.p., n.d. Web. 5 Dec. 2015.		:Description of Motion. N.p., n.d. Web. 5 Dec. 2015.
	:"Speed and Velocity." Speed and Velocity. The Physics Classroom, n.d. Web. 05 Dec. 2015.		:"Speed and Velocity." Speed and Velocity. The Physics Classroom, n.d. Web. 05 Dec. 2015.
			:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Gwang307 at 03:17, 28 November 2016

2016-11-28T03:17:06Z

← Older revision		Revision as of 23:17, 27 November 2016
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	:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.		:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

	:If you ~~simply want to simply see what to derive to get~~ the ~~basic, average velocity~~ formula, <math> {\frac{~~v_{ix}~~ + ~~v_{fx}~~}{2}} </math>, ~~you can~~ do:		:Another way to look at it is where you start from the formula, <math> \frac{v_f + v_i}{2} </math>, and do:

	:<math> v_{avg} = {\frac{\Delta x}{\Delta t}} = {\frac{1}{t_f-t_i}}\int_{t_i}^{t_f}v(t), dt </math>		:<math> v_{avg} = {\frac{\Delta x}{\Delta t}} = {\frac{1}{t_f-t_i}}\int_{t_i}^{t_f}v(t), dt </math>, where <math> v(t) = a(t - t_i) + v_0 </math>
	:where, <math> v(t) = a(t - t_i) + v_0 </math>		:Plugging in <math> v(t) </math> into the integral would result in:
	:Plugging in <math> v(t) </math> into the integral would result in <math> {\frac{1}{t_f-t_i}}\int_{t_i}^{t_f}a(t - t_i) + v_0, dt </math>		:<math> {\frac{1}{t_f-t_i}}\int_{t_i}^{t_f}a(t - t_i) + v_0, dt </math>
			:Integrating separately,
			:<math> \frac{1}{t_f-t_i} * \int a(t_f - t_i) + v_0 - \int a(t_i - t_i) </math>
			:<math>= \frac{1}{t_f-t_i} * \int v_0 + \int at_f - at_i - \int v_0 </math>
			:<math>= \frac{1}{t_f-t_i} * \int at_f - at_i </math>
			:<math>= \frac{1}{t_f-t_i} * \frac{at_f^2}{2} - \frac{at_i^2}{2} </math>
			:<math>= \frac{1}{t_f-t_i} * \frac{at_f^2at_i^2}{2}, (t_f^2-t_i^2) = (t_f-t_i)(t_f+t-i) </math>
			:<math>= \frac{1}{t_f-t_i} * \frac{a(t_f-t_i)(t_f+t_i)}{2} </math>
			:<math>= \frac{a(t_f+t_i)}{2}, a = \frac{v(t)}{t} </math>
			:<math>= \frac{v_f+v_i}{2} </math>
			:Solving the integral results in the original equation, <math> \frac{v_f + v_i}{2} </math>

	===A Computational Model===		===A Computational Model===