Physics Book - User contributions [en]

Derivation of Average Velocity

2016-11-28T04:58:18Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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>

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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. 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===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html
http://www.luc.edu/faculty/dslavsk/courses/ntsc395/classnotes/ntsc395equations.pdf

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.
:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Derivation of Average Velocity

2016-11-28T04:58:11Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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>

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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. 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===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html
http://www.luc.edu/faculty/dslavsk/courses/ntsc395/classnotes/ntsc395equations.pdf

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.
:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Derivation of Average Velocity

2016-11-28T04:58:03Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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>

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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. 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===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html
http://www.luc.edu/faculty/dslavsk/courses/ntsc395/classnotes/ntsc395equations.pdf

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.
:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Derivation of Average Velocity

2016-11-28T04:57:52Z

Gwang307:

: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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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>

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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. 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===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html
http://www.luc.edu/faculty/dslavsk/courses/ntsc395/classnotes/ntsc395equations.pdf

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.
:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Derivation of Average Velocity

2016-11-28T04:57:19Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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>

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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. 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===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html
http://www.luc.edu/faculty/dslavsk/courses/ntsc395/classnotes/ntsc395equations.pdf

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.
:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Derivation of Average Velocity

2016-11-28T03:23:11Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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>

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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.

===External links===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.
:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Derivation of Average Velocity

2016-11-28T03:22:41Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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>

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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.

===External links===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.
:"Derivation Of The Average Velocity Formula With Constant Acceleration (Using Calculus)". Physics.stackexchange.com. N.p., 2016. Web. 27 Nov. 2016.

Derivation of Average Velocity

2016-11-28T03:17:06Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

: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>, where <math> v(t) = a(t - t_i) + v_0 </math>
: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>
: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===

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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.

===External links===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.

Derivation of Average Velocity

2016-11-28T02:35:36Z

Gwang307:

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 is to provide proof of the universal equation for average velocity using derivation. The purpose is also to validate the derivation with fundamental concepts in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x_f - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
: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:

:<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>
: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>

===A Computational Model===

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <math> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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.

===External links===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.

Derivation of Average Velocity

2016-11-28T01:32:33Z

Gwang307:

Derivation of Average Velocity

2016-11-28T00:53:33Z

Gwang307:

Claimed by Gahan Wang (Fall 2016)

==The Main Idea==

The main idea is to provide proof of the universal equation for average velocity. It is also to validate the equation with fundamental concepts and variables both in science and in math.

===A Mathematical Model===

:<math> T </math> = Time
:<math> p </math> = Momentum

'''Geometric Derivation'''

:The equation for average velocity is <math> v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> when velocity in any direction is changing at a constant rate.

:When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. <math> A_{trap} = {\frac{top + bottom}{2}} * altitude = x_f - x_i = {\frac{v_{ix} + v_{fx}}{2}} * (T_f - T_i) </math>

:By dividing the change in time, we get the widely recognized formula for average velocity, <math> {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} </math>

'''Algebraic Derivation'''

:Change in momentum is <math> \Delta p = F_{net} * \Delta t </math> which is also equal to <math> F_{net} = {\frac{\Delta p}{\Delta t}} </math>.
:When evaluating the change in momentum as time approaches zero, <math> F_{net} </math> becomes constant. When the change in time with respect to momentum is 0, <math> p = p_i </math>.

:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>

:<math> v_{avg} = {\frac{x - x_i}{t}} = {\frac{1}{2}}{\frac{F_{net}}{m}}t + v_{ix} = {\frac{1}{2}}(v_{fx} - v_{ix}) + v_{ix} </math>
:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
:After simplifying, <math> V_{avg} = {\frac{v_{ix} + v_{fx}}{2}} </math> where <math> v_x </math> changes at a constant rate.

==Examples==

'''Geometric Model Example'''
:[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = <math> x_{tot} </math>
*Altitude = <nath> \Delta t </math>
*Top side of trapezoid = <math> v_{xi} </math>
*Bottom side of trapezoid = <math> v_{xf} </math>

'''Algebraic Model Example'''
:[[File:Avgvelocity.gif]]

==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.

===External links===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.

Derivation of Average Velocity

2016-11-27T23:48:52Z

Gwang307:

Claimed by Gahan Wang (Fall 2016)

==The Main Idea==

The main idea is to provide proof of the universal equation for average velocity. It is also to validate the equation with fundamental concepts and variables both in science and in math.

===A Mathematical Model===

:T=Time
:P=momentum

'''Geometric Derivation'''

:The equation for average velocity is Vavg= (Vix + Vfx)/2 when velocity in the x direction is changing at a constant rate. The same concept applies to velocity in the y and z directions.

When using geometry as proof, the area of a trapezoid can be used to support the derivation of average velocity. Area of a trapezoid = ((top + bottom)/2) * (altitude) = Xf-Xi= ((Vix + Vfx)/2) * (Tf-Ti)

:By dividing the change in time, we get the widely recognized formula for average velocity, (change in position)/(change in time) = (Vix + Vfx)/2

'''Algebraic Derivation'''

:change in momentum <math> \begin{align} \Delta P \end{align} </math>= net force Fnet * change in time <math> \begin{align} \Delta T \end{align} </math> which is also equal to <math> \begin{align} \Delta P \end{align} </math>/<math> \begin{align} \Delta T \end{align} </math> = Fnet.
:When evaluating the change in momentum as time approaches zero, Fnet becomes constant. When the change in time with respect to momentum is 0, P=Pi.

:Vx=dx/dt=(Fnet/m)t + Vix

:Vavg=(X-Xi)/t=(1/2)(Fnet/m)t + Vix=(1/2)(Vfx-Vix)+Vix
:Vfx=Vx=(Fnet/m)t + Vix
:After simplying, Vavg=(Vix+Vfx)/2 where Vx changes at a constant rate.

==Examples==

Geometric model Example
[[File:TrapezoidExample.jpg]]
*Area of the trapezoid = total displacement
*Altitude = change in time
*Top side of trapezoid=Vxi
*Bottom side of trapezoid=Vxf

Algebraic model Example
*[[File:Avgvelocity.gif]]

==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.

===External links===

:http://physics.tutorvista.com/motion/average-velocity.html
:http://www.mathopenref.com/trapezoidarea.html
:http://www.physicsclassroom.com/class/1DKin/Lesson-1/Speed-and-Velocity
:http://hyperphysics.phy-astr.gsu.edu/hbase/mot.html

==References==

:"Area of a Trapezoid. Definition and Formula - Math Open Reference." Area of a Trapezoid. Definition and Formula - Math Open Reference. Math Open Reference, n.d. Web. 05 Dec. 2015.
:"Average Velocity." Average Velocity. TutorVista, n.d. Web. 05 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.