Derivation of Average Velocity: Difference between revisions

Latest revision as of 17:35, 8 August 2019

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]\displaystyle{ T }[/math] = Time

[math]\displaystyle{ p }[/math] = Momentum

Geometric Derivation

The equation for average velocity is [math]\displaystyle{ 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]\displaystyle{ 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]\displaystyle{ {\frac{\Delta x}{\Delta t}} = {\frac{v_{ix} + v_{fx}}{2}} }[/math]

Algebraic Derivation

Change in momentum is [math]\displaystyle{ \Delta p = F_{net} * \Delta t }[/math] which is also equal to [math]\displaystyle{ F_{net} = {\frac{\Delta p}{\Delta t}} }[/math].

When evaluating the change in momentum as time approaches zero, [math]\displaystyle{ F_{net} }[/math] becomes constant. When the change in time with respect to momentum is 0, [math]\displaystyle{ p = p_i }[/math].

[math]\displaystyle{ v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} }[/math]

[math]\displaystyle{ 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]\displaystyle{ v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} }[/math]

After simplifying, [math]\displaystyle{ v_{avg} = {\frac{v_{ix} + v_{fx}}{2}} }[/math] where [math]\displaystyle{ v_x }[/math] changes at a constant rate.

Another way to look at it is where you start from the formula, [math]\displaystyle{ \frac{v_f + v_i}{2} }[/math], and do:

[math]\displaystyle{ v_{avg} = {\frac{\Delta x}{\Delta t}} = {\frac{1}{t_f-t_i}}\int_{t_i}^{t_f}v(t), dt }[/math], where [math]\displaystyle{ v(t) = a(t - t_i) + v_0 }[/math]

Plugging in [math]\displaystyle{ v(t) }[/math] into the integral would result in:

[math]\displaystyle{ {\frac{1}{t_f-t_i}}\int_{t_i}^{t_f}a(t - t_i) + v_0, dt }[/math]

Integrating separately,

[math]\displaystyle{ \frac{1}{t_f-t_i} * \int a(t_f - t_i) + v_0 - \int a(t_i - t_i) }[/math]

[math]\displaystyle{ = \frac{1}{t_f-t_i} * \int v_0 + \int at_f - at_i - \int v_0 }[/math]

[math]\displaystyle{ = \frac{1}{t_f-t_i} * \int at_f - at_i }[/math]

[math]\displaystyle{ = \frac{1}{t_f-t_i} * \frac{at_f^2}{2} - \frac{at_i^2}{2} }[/math]

[math]\displaystyle{ = \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]\displaystyle{ = \frac{1}{t_f-t_i} * \frac{a(t_f-t_i)(t_f+t_i)}{2} }[/math]

[math]\displaystyle{ = \frac{a(t_f+t_i)}{2}, a = \frac{v(t)}{t} }[/math]

[math]\displaystyle{ = \frac{v_f+v_i}{2} }[/math]

Solving the integral results in the original equation, [math]\displaystyle{ \frac{v_f + v_i}{2} }[/math]

Examples

Geometric Model Example

Area of the trapezoid = [math]\displaystyle{ x_{tot} }[/math]
Altitude = [math]\displaystyle{ \Delta t }[/math]
Top side of trapezoid = [math]\displaystyle{ v_{xi} }[/math]
Bottom side of trapezoid = [math]\displaystyle{ v_{xf} }[/math]

Algebraic Model Example

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.

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.

@@ Line 1: / Line 1: @@
 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. It is also to validate the equation with fundamental concepts and variables both in science and in math.
+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===
-:T=Time
+:<math> T </math> = Time
-:P=momentum
+:<math> p </math> = 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.
+: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. Area of a trapezoid = ((top + bottom)/2) * (altitude) = Xf-Xi= ((Vix + Vfx)/2) * (Tf-Ti)
+: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, (change in position)/(change in time) = (Vix + Vfx)/2
+: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 <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.
+: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, Fnet becomes constant. When the change in time with respect to momentum is 0, P=Pi.
+: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>.
-:Vx=dx/dt=(Fnet/m)t + Vix
+:<math> v_x = {\frac{dx}{dt}} = {\frac{F_{net}}{m}}t + v_{ix} </math>
-:Vavg=(X-Xi)/t=(1/2)(Fnet/m)t + Vix=(1/2)(Vfx-Vix)+Vix
+:<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>
-:Vfx=Vx=(Fnet/m)t + Vix
+:<math> v_{fx} = v_x = {\frac{F_{net}}{m}}t + v_{ix} </math>
-:After simplying, Vavg=(Vix+Vfx)/2 where Vx 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.
+: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
+'''Geometric Model Example'''
-[[File:TrapezoidExample.jpg]]
+:[[File:TrapezoidExample.jpg]]
-*Area of the trapezoid = total displacement
+*Area of the trapezoid = <math> x_{tot} </math>
-*Altitude = change in time
+*Altitude = <math> \Delta t </math>
-*Top side of trapezoid=Vxi
+*Top side of trapezoid = <math> v_{xi} </math>
-*Bottom side of trapezoid=Vxf
+*Bottom side of trapezoid = <math> v_{xf} </math>
-Algebraic model Example
+'''Algebraic Model Example'''
-*[[File:Avgvelocity.gif]]
+:[[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.
+: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===
@@ Line 52: / Line 74: @@
 :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==
@@ Line 59: / Line 82: @@
 :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.