# Connected Chemistry 3 Circular Particles

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## WHAT IS IT?

This model explores the relationship between particle kinetic energies during particle collisions. This model is part of the "Connected Chemistry" curriculum http://ccl.northwestern.edu/curriculum/ConnectedChemistry/ which explore the behavior of gases.

This model is different from the other Connected Chemistry and GasLab models in that the collision calculations take into account the circular shape and size of the particles, instead of modeling the particles as dimensionless points.

## HOW IT WORKS

The model determines the resulting motion of particles that collide, with no loss in their total momentum or total kinetic energy (an elastic collision).

To calculate the outcome of collision, it is necessary to calculate the exact time at which the edge of one particle (represented as a circle), would touch the edge of another particle (or the walls of a container) if the particles were allowed to continue with their current headings and speeds.

By performing such a calculation, one can determine when the next collision anywhere in the system would occur in time. From this determination, the model then advances the motion of all the particles using their current headings and speeds that far in time until this next collision point is reached. Exchange of kinetic energy and momentum between the two particles, according to conservation of kinetic energy and conservation of momentum along the collision axis (a line drawn between the centers of the two particles), is then calculated, and the particles are given new headings and speeds based on this outcome.

## HOW TO USE IT

INITIAL-NUMBER determines the number of gas particles used with SETUP. If the world is too small or the particles are too large, the SETUP procedure of the particles will stop so as to prevent overlapping particles.

Choosers: SHOW-SPEED-AS-COLOR? allows you to visualize particle speed using a color palette.

- The "blue-green-red" setting shows the lower half of the speeds of the starting population as blue, and the upper half as red.
- The "violet shades" setting shows a gradient from dark violet (slow) to light violet (fast).
- The "all green" setting shows all particles in green, regardless of speed.
- The "custom color" setting, referenced in the Pedagogica version of this model, allows the user to modify the color of one or more particles, without having to worry that the particles will be recolored with each tick of the clock (as is the case for the other color options).

Monitors: CLOCK - number of clock cycles that GO has run. PARTICLE 0 SPEED - reports the speed for one of the particles (the first particle created that has a who value of 0) PARTICLE 0 HEADING - reports the heading for one of the particles (the first particle created that has a who value of 0)

Plots: KINETIC ENERGY VS TIME - plots the kinetic energy for two particles. If the model is run with only two particles for the INITIAL-NUMBER slider, then this graph will show that kinetic energy is conserved in collisions between these two particles. PARTICLE 0 SPEED VS. TIME - plots the speed for one of the particles (the first particle created that has a who value of 0) over time. PARTICLE 0 HEADING - plots the heading for one of the particles (the first particle created that has a who value of 0) over time.

## THINGS TO NOTICE

Particles never overlap or penetrate into each other or the wall as they move about.

For only two particles in the world, what do you notice about the kinetic energy of the particles before and after a collision?

Does kinetic energy of a particle (e.g. particle 0) change when a particle hits the wall?

## THINGS TO TRY

Setting all the particles to have a very slow speed (e.g. 0.001) and one particle to have a very fast speed helps show how kinetic energy is eventually transferred to all the particles through a series of collisions and would serve as a good model for energy exchange through conduction between hot and cold gases.

To see what the approximate mass of each particle is, type this in the command center: ask particles [set label precision mass 0]

## THINGS TO TRY

Use the speed slider (at the top of the View) to run the model in slow motion.

Use the command center to set the speed of all the particles to a very low number (but not zero - e.g. 0.01) and set the speed of one particle to a very high number (e.g. 1000). Watch how the kinetic energy of the fast particle is distributed in collisions to the other particles.

Use the command center to trace the path of a particle, using pendown.

How is kinetic energy represented by color when you set the SHOW-SPEED-AS-COLOR? chooser to "violet shades"?

## EXTENDING THE MODEL

Collisions between boxes and circles could also be explored. Variations in size between particles could investigated or variations in the mass of some of the particle could be made to explore other factors that affect the outcome of collisions.

## NETLOGO FEATURES

The global variable VIEW-UPDATE-RATE determines the minimum number of simulation clock ticks that must pass before the view is updated. This helps to smooth out the motion of the particles seen in the View. The particles should appear to have a linear rate of change in their motion when they are not colliding. Without a minimum VIEW-UPDATE-RATE, however, this would not appear to be the case, because the model would update the graphics after every tick, and ticks vary in length, depending on how close the next projected collision is. As the tick length moved up and down, the simulation time the particles were allowed to move forward before they were redisplayed would also move up and down. This would cause the motion of the particles to look jerky and non-linear.

## RELATED MODELS

See GasLab Circular Particles See other Connected Chemistry models.

## CREDITS AND REFERENCES

This model is part of the Connected Chemistry curriculum. See http://ccl.northwestern.edu/curriculum/chemistry.

We would like to thank Sharona Levy and Michael Novak for their substantial contributions to this model.

## HOW TO CITE

If you mention this model in a publication, we ask that you include these citations for the model itself and for the NetLogo software:

- Wilensky, U. (2005). NetLogo Connected Chemistry 3 Circular Particles model. http://ccl.northwestern.edu/netlogo/models/ConnectedChemistry3CircularParticles. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.
- Wilensky, U. (1999). NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

To cite the Connected Chemistry curriculum as a whole, please use: Wilensky, U., Levy, S. T., & Novak, M. (2004). Connected Chemistry curriculum. http://ccl.northwestern.edu/curriculum/chemistry. Center for Connected Learning and Computer-Based Modeling, Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

Copyright 2005 Uri Wilensky.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

Commercial licenses are also available. To inquire about commercial licenses, please contact Uri Wilensky at uri@northwestern.edu.

## Comments and Questions

globals [ tick-delta ;; how much we advance the tick counter this time through max-tick-delta ;; the largest tick-delta is allowed to be box-edge ;; distance of box edge from axes colliding-particles sorted-colliding-particles colliding-particle-1 colliding-particle-2 colliding-pair original-tick-delta last-view-update manage-view-updates? view-update-rate ;; specifies the minimum amount of simulation time that must ;; pass before the view is updated winners smallest-particle-size largest-particle-size kinetic-energy-of-one kinetic-energy-of-two pressure total-particle-number total-energy ] breed [ particles particle ] particles-own [ speed mass ] ;;;;;;;;;;;;;;;; to setup clear-all reset-ticks set total-particle-number initial-number set-default-shape particles "circle" set manage-view-updates? true set view-update-rate 0.2 set box-edge (max-pxcor - 1) set smallest-particle-size 4 set largest-particle-size 4 set kinetic-energy-of-one 0 set kinetic-energy-of-two 0 make-box make-particles ;; set variable tick delta based on fastest particle. If the fastest particle has a speed of 1, ;; then tick-delta is 1. If the fastest particles has a speed of 10, then tick-delta is 1/10. set tick-delta (1 / (ceiling max [speed] of particles)) set original-tick-delta tick-delta set colliding-particles [] ask particles [check-for-wall-collision] ask particles [check-for-particle-collision ] set colliding-particle-1 nobody set colliding-particle-2 nobody ifelse labels? [ask turtles [ set label who set label-color orange + 3 ]] [ask turtles [ set label ""]] end to go ;;Since only collisions involving the particles that collided most recently could be affected, ;;we filter those out of colliding-particles. Then we recalculate all possible collisions for the ;;particles that collided last. The ifelse statement is necessary because colliding-particle-2 ;;can be either a particle or a string representing a wall. If it is a wall, we don't want to ;;invalidate all collisions involving that wall (because the wall's position wasn't affected, those ;;collisions are still valid. ifelse is-turtle? colliding-particle-2 [ set colliding-particles filter [item 1 ? != colliding-particle-1 and item 2 ? != colliding-particle-1 and item 1 ? != colliding-particle-2 and item 2 ? != colliding-particle-2] colliding-particles ask colliding-particle-2 [check-for-wall-collision] ask colliding-particle-2 [check-for-particle-collision] ] [ set colliding-particles filter [item 1 ? != colliding-particle-1 and item 2 ? != colliding-particle-1] colliding-particles ] if colliding-particle-1 != nobody [ask colliding-particle-1 [check-for-wall-collision]] if colliding-particle-1 != nobody [ask colliding-particle-1 [check-for-particle-collision]] sort-collisions calculate-energies ask particles [ jump speed * tick-delta ] if winners != [] [collide-winners] tick-advance tick-delta do-plots ask particles [ if show-speed-as-color? = "red-green-blue" [ recolor-banded ] if show-speed-as-color? = "violet shades" [ recolor-shaded ] if show-speed-as-color? = "one color" [ recolor-none ] if show-speed-as-color? = "custom color" [ ] ] ifelse labels? [ask turtles [ set label who set label-color orange + 3 ]] [ask turtles [ set label ""]] ;; flag that updates display only after enough simulation time has passed. ;; the display-update-rate sets the minimum simulation time that must pass ;; before updating the display. This avoids many redisplays of the view for ;; a series of small time steps in the simulation (which would make the view show ;; what looks like particles slowing down as they get near multiple collision points) if manage-view-updates? [ if (ticks - last-view-update) > view-update-rate [ display set last-view-update ticks ] ] end to calculate-energies ask particle 0 [ set kinetic-energy-of-one ( 0.5 * mass * (speed * speed)) ] ask particle 1 [ set kinetic-energy-of-two ( 0.5 * mass * (speed * speed)) ] end to-report convert-heading-x [heading-angle] report sin heading-angle end to-report convert-heading-y [heading-angle] report cos heading-angle end to-report future-time? [time] ;;This reporter is necessary because of slight discrepancies in the floating point math. ;;Sometimes particles that are colliding at that instant are reported as colliding again ;;imperceptibly in the future; this causes the model to hang. ;;This function ensures that all expected collisions are legitimate collisions, and not just ;;ghosts of floating point errors. report time > .0000000001 end to check-for-particle-collision ;; check-for-particle-collision is a particle procedure that determines the time it takes to the collision between ;; two particles (if one exists). It solves for the time by representing the equations of motion for ;; distance, velocity, and time in a quadratic equation of the vector components of the relative velocities ;; and changes in position between the two particles and solves for the time until the next collision let my-x xcor let my-y ycor let my-particle-size size let my-x-speed (speed * convert-heading-x heading) let my-y-speed (speed * convert-heading-y heading) ask particles with [self != myself] [ let dpx (xcor - my-x) ;; relative distance between particles in the x direction let dpy (ycor - my-y) ;; relative distance between particles in the y direction let x-speed (speed * convert-heading-x heading) ;; speed of other particle in the x direction let y-speed (speed * convert-heading-y heading) ;; speed of other particle in the x direction let dvx (x-speed - my-x-speed) ;; relative speed difference between particles in the x direction let dvy (y-speed - my-y-speed) ;; relative speed difference between particles in the y direction let sum-r (((my-particle-size) / 2 ) + (([size] of self) / 2 )) ;; sum of both particle radii ;; To figure out what the difference in position (P1) between two particles at a future time (t) would be, ;; one would need to know the current difference in position (P0) between the two particles ;; and the current difference in the velocity (V0) between of the two particles. ;; The equation that represents the relationship would be: P1 = P0 + t * V0 ;; we want find when in time (t), P1 would be equal to the sum of both the particle's radii (sum-r). ;; When P1 is equal to is equal to sum-r, the particles will just be touching each other at ;; their edges (a single point of contact). ;; Therefore we are looking for when: sum-r = P0 + t * V0 ;; This equation is not a simple linear equation, since P0 and V0 should both have x and y components ;; in their two dimensional vector representation (calculated as dpx, dpy, and dvx, dvy). ;; By squaring both sides of the equation, we get: (sum-r) * (sum-r) = (P0 + t * V0) * (P0 + t * V0) ;; When expanded gives: (sum-r ^ 2) = (P0 ^ 2) + (t * PO * V0) + (t * PO * V0) + (t ^ 2 * VO ^ 2) ;; Which can be simplified to: 0 = (P0 ^ 2) - (sum-r ^ 2) + (2 * PO * V0) * t + (VO ^ 2) * t ^ 2 ;; Below, we will let p-squared represent: (P0 ^ 2) - (sum-r ^ 2) ;; and pv represent: (2 * PO * V0) ;; and v-squared represent: (VO ^ 2) ;; then the equation will simplify to: 0 = p-squared + pv * t + v-squared * t^2 let p-squared ((dpx * dpx) + (dpy * dpy)) - (sum-r ^ 2) ;; p-squared represents difference of the ;; square of the radii and the square ;; of the initial positions let pv (2 * ((dpx * dvx) + (dpy * dvy))) ;;the vector product of the position times the velocity let v-squared ((dvx * dvx) + (dvy * dvy)) ;; the square of the difference in speeds ;; represented as the sum of the squares of the x-component ;; and y-component of relative speeds between the two particles ;; p-squared, pv, and v-squared are coefficients in the quadratic equation shown above that ;; represents how distance between the particles and relative velocity are related to the time, ;; t, at which they will next collide (or when their edges will just be touching) ;; Any quadratic equation that is the function of time (t), can represented in a general form as: ;; a*t*t + b*t + c = 0, ;; where a, b, and c are the coefficients of the three different terms, and has solutions for t ;; that can be found by using the quadratic formula. The quadratic formula states that if a is not 0, ;; then there are two solutions for t, either real or complex. ;; t is equal to (b +/- sqrt (b^2 - 4*a*c)) / 2*a ;; the portion of this equation that is under a square root is referred to here ;; as the determinant, D1. D1 is equal to (b^2 - 4*a*c) ;; and: a = v-squared, b = pv, and c = p-squared. let D1 pv ^ 2 - (4 * v-squared * p-squared) ;; the next line next line tells us that a collision will happen in the future if ;; the determinant, D1 is >= 0, since a positive determinant tells us that there is a ;; real solution for the quadratic equation. Quadratic equations can have solutions ;; that are not real (they are square roots of negative numbers). These are referred ;; to as imaginary numbers and for many real world systems that the equations represent ;; are not real world states the system can actually end up in. ;; Once we determine that a real solution exists, we want to take only one of the two ;; possible solutions to the quadratic equation, namely the smaller of the two the solutions: ;; (b - sqrt (b^2 - 4*a*c)) / 2*a ;; which is a solution that represents when the particles first touching on their edges. ;; instead of (b + sqrt (b^2 - 4*a*c)) / 2*a ;; which is a solution that represents a time after the particles have penetrated ;; and are coming back out of each other and when they are just touching on their edges. let time-to-collision -1 if D1 >= 0 and v-squared > 0 [set time-to-collision (- pv - sqrt D1) / (2 * v-squared) ] ;;solution for time step ;; if time-to-collision is still -1 there is no collision in the future - no valid solution ;; note: negative values for time-to-collision represent where particles would collide ;; if allowed to move backward in time. ;; if time-to-collision is greater than 1, then we continue to advance the motion ;; of the particles along their current trajectories. They do not collide yet. if future-time? time-to-collision [ ;; time-to-collision is relative (ie, a collision will occur one second from now) ;; We need to store the absolute time (ie, a collision will occur at time 48.5 seconds. ;; So, we add ticks to time-to-collision when we store it. set colliding-pair (list (time-to-collision + ticks) self myself) ;; sets a three element list of ;; time to collision and the colliding pair set colliding-particles lput colliding-pair colliding-particles ;; adds above list to collection ;; of colliding pairs and time ;; steps ] ] end to check-for-wall-collision ;; particle procedure for determining if a particle will hit one of the ;; four walls of the box let x-speed (speed * convert-heading-x heading) let y-speed (speed * convert-heading-y heading) let xpos-plane (box-edge - 0.5) ;;inside boundary of right side of the box let xneg-plane (- box-edge + 0.5) ;;inside boundary of left side of the box let ypos-plane (box-edge - 0.5) ;;inside boundary of top side of the box let yneg-plane (- box-edge + 0.5) ;;inside boundary of bottom side of the box ;; find point of contact on edge of circle ;; points of contact located at 1 radius above, below, to the left, and to the right ;; of the center of the particle let contact-point-xpos (xcor + (size / 2)) let contact-point-xneg (xcor - (size / 2)) let contact-point-ypos (ycor + (size / 2)) let contact-point-yneg (ycor - (size / 2)) ;; find difference in position between plane location and edge of circle let dpxpos (xpos-plane - contact-point-xpos) let dpxneg (xneg-plane - contact-point-xneg) let dpypos (ypos-plane - contact-point-ypos) let dpyneg (yneg-plane - contact-point-yneg) let t-plane-xpos 0 ;; solve for the time it will take the particle to reach the wall by taking ;; the distance to the wall and dividing it by the speed in the direction to the wall ifelse x-speed != 0 [set t-plane-xpos (dpxpos / x-speed)] [set t-plane-xpos 0] if future-time? t-plane-xpos [ assign-colliding-wall t-plane-xpos "plane-xpos" ] let t-plane-xneg 0 ifelse x-speed != 0 [set t-plane-xneg (dpxneg / x-speed)] [set t-plane-xneg 0] if future-time? t-plane-xneg [ assign-colliding-wall t-plane-xneg "plane-xneg" ] let t-plane-ypos 0 ifelse y-speed != 0 [set t-plane-ypos (dpypos / y-speed)] [set t-plane-ypos 0] if future-time? t-plane-ypos [ assign-colliding-wall t-plane-ypos "plane-ypos" ] let t-plane-yneg 0 ifelse y-speed != 0 [set t-plane-yneg (dpyneg / y-speed)] [set t-plane-yneg 0] if future-time? t-plane-yneg [ assign-colliding-wall t-plane-yneg "plane-yneg" ] end to assign-colliding-wall [time-to-collision wall] ;; this procedure is used by the check-for-wall-collision procedure ;; to assemble the correct particle-wall pair ;; time-to-collision is relative (ie, a collision will occur one second from now) ;; We need to store the absolute time (ie, a collision will occur at time 48.5 seconds. ;; So, we add ticks to time-to-collision when we store it. set colliding-pair (list (time-to-collision + ticks) self wall) set colliding-particles lput colliding-pair colliding-particles end to sort-collisions ;; Sort the list of projected collisions between all the particles into an ordered list. ;; Take the smallest time-step from the list (which represents the next collision that will ;; happen in time). Use this time step as the tick-delta for all the particles to move through ifelse colliding-particles != [] [ set winners first colliding-particles foreach colliding-particles [if first ? < first winners [set winners ?]] ;;winners is now the collision that will occur next let dt first winners set colliding-particle-1 item 1 winners set colliding-particle-2 item 2 winners if dt > 0 [ ;;If the next collision is more than 1 in the future, ;;clear the winners and advance the simulation one tick. ;;This helps smooth the model on smaller particle counts. ifelse dt - ticks <= 1 ;;We have to subtract ticks back out because now we want the relative time until collision, ;;not the absolute time the collision will occur. [set tick-delta dt - ticks] ;;Since there are no collisions in the next second, we will set winners to [] to keep from ;;mistakenly colliding any particles that shouldn't collide yet. [set tick-delta 1 set winners []] ] ] ;; When there are no collisions for the next time step, ;; tick-delta goes back to the value of original-tick-delta [set tick-delta original-tick-delta] end to collide-winners ;; deal with 3 possible cases of collisions: ;; particle and one wall, particle and two walls, and two particles ;; deal with a case where the next collision in time is between a particle and a wall if colliding-particle-2 = "plane-xpos" or colliding-particle-2 = "plane-xneg" [ask colliding-particle-1 [set heading (- heading)] stop] if colliding-particle-2 = "plane-ypos" or colliding-particle-2 = "plane-yneg" [ask colliding-particle-1 [set heading (180 - heading)] stop] ;; deal with the remaining case of the next collision in time being between two particles. ask colliding-particle-1 [collide-with colliding-particle-2] end to collide-with [ other-particle ] ;; particle procedure ;;; PHASE 1: initial setup ;; for convenience, grab some quantities from other-particle let mass2 [mass] of other-particle let speed2 [speed] of other-particle let heading2 [heading] of other-particle ;;modified so that theta is heading toward other particle let theta towards other-particle ;;; PHASE 2: convert velocities to theta-based vector representation ;; now convert my velocity from speed/heading representation to components ;; along theta and perpendicular to theta let v1t (speed * cos (theta - heading)) let v1l (speed * sin (theta - heading)) ;; do the same for other-particle let v2t (speed2 * cos (theta - heading2)) let v2l (speed2 * sin (theta - heading2)) ;;; PHASE 3: manipulate vectors to implement collision ;; compute the velocity of the system's center of mass along theta let vcm (((mass * v1t) + (mass2 * v2t)) / (mass + mass2) ) ;; now compute the new velocity for each particle along direction theta. ;; velocity perpendicular to theta is unaffected by a collision along theta, ;; so the next two lines actually implement the collision itself, in the ;; sense that the effects of the collision are exactly the following changes ;; in particle velocity. set v1t (2 * vcm - v1t) set v2t (2 * vcm - v2t) ;;; PHASE 4: convert back to normal speed/heading ;; now convert my velocity vector into my new speed and heading set speed sqrt ((v1t * v1t) + (v1l * v1l)) ;; if the magnitude of the velocity vector is 0, atan is undefined. but ;; speed will be 0, so heading is irrelevant anyway. therefore, in that ;; case we'll just leave it unmodified. if v1l != 0 or v1t != 0 [ set heading (theta - (atan v1l v1t)) ] ;; and do the same for other-particle ask other-particle [ set speed sqrt ((v2t ^ 2) + (v2l ^ 2)) if v2l != 0 or v2t != 0 [ set heading (theta - (atan v2l v2t)) ] ] ;; PHASE 5: final updates ;; now recolor, since color is based on quantities that may have changed recolor ask other-particle [ recolor ] end to recolor if show-speed-as-color? = "red-green-blue" [ recolor-banded ] if show-speed-as-color? = "violet shades" [ recolor-shaded ] if show-speed-as-color? = "one color" [ recolor-none ] end to recolor-banded ;; particle procedure let avg-speed 1 ;; avg-speed is assumed to be 0.5, since particles are assigned a random speed between 0 and 1 ;; particle coloring procedures for visualizing speed with a color palette, ;; red are fast particles, blue slow, and green in between. ifelse speed < (0.5 * avg-speed) ;; at lower than 50% the average speed [ set color blue ;; slow particles colored blue ] [ ifelse speed > (1.5 * avg-speed) ;; above 50% higher the average speed [ set color red ] ;; fast particles colored blue [ set color green ] ;; medium speed particles colored green ] end to recolor-shaded let avg-speed 1 ;; avg-speed is assumed to be 0.5, since particles are assigned a random speed between 0 and 1 ;; a particle shading gradient is applied to all particles less than speed 1.5, ;; the uppermost threshold speed to apply the shading gradient to. ifelse speed < (3 * avg-speed) [ set color (violet - 3.001) + (8 * speed / (3 * avg-speed)) ] [ set color (violet + 4.999)] end to recolor-none set color green - 1 end ;;; ;;; drawing procedures ;;; to make-box ask patches with [ ((abs pxcor = box-edge) and (abs pycor <= box-edge)) or ((abs pycor = box-edge) and (abs pxcor <= box-edge)) ] [ set pcolor yellow ] end ;; creates some particles to make-particles create-particles initial-number [ set speed 1 set size smallest-particle-size + random-float (largest-particle-size - smallest-particle-size) set mass (size * size) ;; set the mass proportional to the area of the particle random-position recolor ] end ;; place particle at random location inside the box. to random-position ;; particle procedure let count-loops 0 let radius (size / 2) let largest-radius (largest-particle-size / 2) let both-radii (radius + largest-radius) ;;this loop only works with < 25 particles while [(count particles in-radius both-radii) > 1 and count-loops < 5000] [ setxy ((radius - (box-edge - size )) + random-float ((2 * (box-edge - size )) - (2 * radius))) ((radius - (box-edge - size )) + random-float ((2 * (box-edge - size )) - (2 * radius))) set count-loops count-loops + 1 ] end to do-plots set-current-plot "Particle 0 Heading vs. Time" plotxy ticks ([heading] of particle 0) set-current-plot "Particle 0 Speed vs. Time" plotxy ticks ([speed] of particle 0) set-current-plot "Kinetic Energy vs. Time" set-current-plot-pen "particle 0" plotxy ticks kinetic-energy-of-one set-current-plot-pen "particle 1" plotxy ticks kinetic-energy-of-two end ; Copyright 2005 Uri Wilensky. ; See Info tab for full copyright and license.

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## Attached files

File | Type | Description | Last updated | |
---|---|---|---|---|

Connected Chemistry 3 Circular Particles.png | preview | Preview | about 6 years ago, by Reuven M. Lerner | Download |

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