ball rolling down a ramp simulation

Adobe Stock. A problem about harmonic oscillators. The AIP Style presented is based on information from the AIP Style Manual. Open content licensed under CC BY-NC-SA, Snapshot 1: the initial position of the ball; the velocity at this time is 0, Snapshot 2: after a time, and at a height, the ball has moved down to its current position, Snapshot 3: after the same time, and at the same height, the ball has moved down to its current position; this position is different from the position of snapshot 2. Try the experiment with different ramp angles. The Graphs and Ramps Interactive is a simulation in which learners build a ramp along which a ball will roll. Just like the bells on Galileo's ramp, the positions of three of the vertical red lines can be adjusted. roll the ball down and measure the time it takes and the distance it travels before it hits the floor. C. Compare the time for the ball to roll from 0 to 50 cm to the time for the ball to roll from 200 cm to 250 cm. The counter has been running on this page since 8-10-2018. Since the perceptual deficiencies have been reported in studies involving a limited visual context, here we tested the hypothesis that judgments of . Biology, 22.06.2019 02:00. N. Mihara, (Wisconsin Society of Science Teachers, Oshkosh, 2000), WWW Document, (. Number = {3 March 2023}, Use the ruler or meter stick to mark 10 cm intervals along the ramp, starting at the floor and going upward. In this eighth-grade geometry worksheet, students practice graphing images of figures after completing translations on a coordinate plane. The cube slides without friction, the other objects roll without slipping. Photos Illustrations Vecteurs Vidos Audio Templates Gratuit Premium Polices. "Effect of Friction on Ball Rolling Down a Ramp" Use suvat equations to work out the speed and acceleration ect of the ball and you can easily work it out. N. Mihara, Ramp n Roll (Wisconsin Society of Science Teachers, Oshkosh, 2000), . Relate this resource If you change the angle of the ramp to be steeper, the acceleration you record will be closer to that of gravity. 1996-2022 The Physics Classroom, All rights reserved. In this Blender tutorial, I show you how to create a rigid body physics simulation of a ball rolling down a ramp and jumping into a cup.- - - - - - - - - - - - - - - - - - - - - - - - - - - -Background Music:(NA) by (NA)Available free to use and monetize.- - - - - - - - -Some of my most popular videos:https://www.youtube.com/watch?v=bb62m02pU2Ahttps://www.youtube.com/watch?v=d-8bLbL3sXQhttps://www.youtube.com/watch?v=guuOAuAIq3Qhttps://www.youtube.com/watch?v=wRjL7Jm2LKIhttps://www.youtube.com/watch?v=jqqrNTDLbSghttps://www.youtube.com/watch?v=WJruYS5qAfwhttps://www.youtube.com/watch?v=J9bSYIbpffMhttps://www.youtube.com/watch?v=Bgab9VC_laQhttps://www.youtube.com/watch?v=pKU9D01cv88https://www.youtube.com/watch?v=qqP5YHbk8eg- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - FIND ME ON SOCIAL MEDIA:Odysee: https://odysee.com/@BlenderRookie:8BlendSwap:https://www.blendswap.com/user/BlenderRookie2Blender Rookie:https://www.youtube.com/c/BlenderRookieSocialbladehttps://socialblade.com/youtube/channel/UC3FTjAoLBC00HHK5cSSL7kA- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - DONATE:Paypalhttps://paypal.me/BlenderRookie- - - - - - - - - - - - - - - - - - - - - - - - - - - -The Software I Use:Live Streaming/Screen recording: OBS(freeware)https://obsproject.com/3D Editor: Blender 3D (freeware)https://www.blender.orgVideo Editor: Cyberlink PowerDirector14 (payware)https://www.cyberlink.comMP4 Encoder: HandBrake (freeware)https://handbrake.frAudio Editor: Audacity (freeware)http://www.audacityteam.org- - - - - - - - - - - - - - - - - - - - - - - - - - - -My Hardware:Desktop 1Motherboard: Asus X570 TUFCPU: Ryzen 9 3900XGPU: RTX 3070 - RTX 3060RAM: 64GB 3600Mhz G Skill Ripjaws 4 X 16GBSTORAGE: 500GB NVME (Western Digital Black) 1TB Samsung Evo NVMECOOLING: Cooler Master Hyper 212POWER SUPPLY: EVGA 850 GoldDesktop 2Motherboard: MSI X470CPU: Ryzen 5 3600GPU: 2X RTX 2060 Super - 2X GTX 1060 6GBRAM: 32GB 2133Mhz G Skill Ripjaws 4 X 8GBSTORAGE: 250GB SSD Western Digital BlueCOOLING: Stock AMD CoolerPOWER SUPPLY: EVGA 500 Bronze- - - - - - - - - - - - - - - - - - - - - - - - - - - -Keywords \u0026 Hashtags:#Satisfying #KevaPlank #BlenderRender #Blender3D #Freeware #Animation #CGI #VFX #ParticleFluids #BlenderRookie - - - - - - - - - - - - - - - - - - - - - - - - - - - -I am a Blender Rookie teaching myself how to use Blender. Answers: 1 Show answers Another question on Biology. This demonstration can also be used to show the static frictioncoefficients of different materials and how the force on an object will increase as the angle of the surface it lies on increases. Record the final angle in your notebook. Lower and raise the ramp to see how the angle of inclination affects the parallel forces acting on the file cabinet. The applet then displays the motion of the ball as well as position, velocity, and acceleration graphs in real time. Use the Run, Pause, and Reset buttons to control the animation, and the speed slider to adjust the animation speed. You can plot the total mechanical energy (purple), gravitational potential energy (red), kinetic energy (green), and the thermal energy (black) as a function of time or position. You can calculatet for each of the four segments of ramp with the equation: t1 = t2 t1 The site also provides drawing tools for users to draw . There are two limiting cases, one with no friction and one with friction, so there is no slippage of the ball. Is there a net gravitional foce at the center of the earth? You can plot the total mechanical energy (purple), gravitational potential energy (red), Caili Chen See a problem with this material's physics or description? Contributed by: Athena Hung and Caili Chen(June 2014) "Special thanks to the University of Illinois NetMath Program and the mathematics department at William Fremd High School." This is a simulation of objects sliding and rolling down an incline. In this simulation, the user can explore the rolling motion of various objects with varying rotational inertia. Instead of dropping an object so that it would free-fall, Galileo timed the motion of balls rolling down ramps. Title = {Ramp n Roll}, Title = {Ramp n Roll}, Have experience with this material? The different mass distributions cause the rolling objects to have different rotational inertia, so they roll down the incline with different . Blender Rookie 24.6K subscribers In this Blender tutorial, I show you how to create a rigid body physics simulation of a ball rolling down a ramp and jumping into a cup. This page: Rolling Motion looks at the situations when the ball is rolling without slipping and when it isn't. Each case, however, gives a different formula for the force imparted by the contact of the ball with the incline. The cube slides without friction, the other objects roll without slipping. In this simulation, the user can explore the rolling motion of various objects with varying rotational inertia. We will surely have to conduct many different experiments. translational kinetic energy (green), and rotational kinetic energy (blue) as a function of time or position. Galileo stated that objects in a vacuum, meaning no air, would fall to the Earth with a constant acceleration. Acceleration due to gravity is measured as 9.81 m/s2. The different mass distributions cause the rolling objects to have different rotational inertia, so they roll down the incline with different . Warning - you are about to disable cookies. Physics 110A & B: Electricity, Magnetism, and Optics (Parts I & II), Physics 112: Thermodynamics and Statistical Mechanics, 50.8 mm diameter steel ball, mass 534.6 g, 2x small clamps to attach protractor to slope, Plump bob/string (thin fishing line and 20g weight, found in blackboard mechanics). This demo is similar to the static and kinetic friction demo, but instead of changing the weight required to make the block move, we can change the angle of the plane. If you would prefer to use the older version, Click here. If yes, then prepare yourself for this highly engaging Rolling Ball: Car Drift Racing. Since the incline of the ramp, the mass of the ball and the value . The applet then displays the motion of the ball as well as position, velocity, and acceleration graphs in real time. The user can set the ball's initial position and velocity and the geometry of the ramp. Rolling - four views; How a front-wheel-drive car works; Rolling - the bowling ball problem; Jumping on a merry-go-round; An accelerating cylinder; Rolling down a ramp; Harmonic Motion. Rescue Mission: Graphing on a Coordinate Plane, Treasure Hunting: Graphing on a Coordinate Plane, Transformations on the Coordinate Plane: Dilations Handout, Transformations on the Coordinate Plane: Rotations Handout, Transformations on the Coordinate Plane: Translations Handout, 3 feet of molding (for a ceiling or floor, with a groove to roll a ball down), Computer with Excel (unless you want to graph by hand!). You will need to take eight different time measurements and will calculate four velocities and two accelerations. This is a simulation of objects sliding and rolling down an incline. Publisher = {Wisconsin Society of Science Teachers}, Ball sliding down a ramp. 3 cm 77 cm 60. You can plot the total mechanical energy (purple), gravitational potential energy (red), translational kinetic energy (green), and rotational kinetic energy (blue) as a function of time or position. In Dilations on the Coordinate Plane, students will practice graphing images of figures after completing given dilations, all of whichare centered at the origin. Make a Comment 50 cm 100 cm. Graph your results. $\begingroup$ x is the horizontal distance between the end of the ramp and where the ball hits the ground. Interact on desktop, mobile and cloud with the free WolframPlayer or other Wolfram Language products. Does the Sun's gravity decrease as it loses mass. Stack some books and set one side of the molding on the books to create a ramp. A ball rolling down a hill: it's not exactly an F1 car zooming round Eau Rouge, but the laws of physics are the same! - - - - - - - - -. http://demonstrations.wolfram.com/EffectOfFrictionOnBallRollingDownARamp/ While the gravitational force acting on the block does not change depending on the angle of the board, a steeper incline will give a larger component force that is pushing the block down the ramp. Login to relate this resource to other material across the web. Horizontal position of bell 2. We enable strictly necessary cookies to give you the best possible experience on Education.com. This will yield V1, V2, V3, V4, which we can use to find two accelerations, a1, a2. What the ramp should look like if marked for constant acceleration demonstration, where the change in x should be equal across all four distances. We need to conduct experiments to find out how changing the angle of the ramp, the length of the ramp, and the mass of the ball affects how far the ball rolls. The Chicago Style presented is based on information from Examples of Chicago-Style Documentation. Today, we call this constant acceleration gravity. This is not realistic at very large angles of incline. . Repeat step for at different lengths along the ramp. What is the kinetic energy in C? The MLA Style presented is based on information from the MLA FAQ. Written by Andrew Duffy. Using that the mechanical energy is the sum of potential energy and kinetic energy , we get that the mechanical energies in are , respectively: They must be equal. ComPADRE is beta testing Citation Styles! The goal is to build the ramp with the correct heights and incline angles such that the roling ball moves with a motion that matches a provided position-time or velocity-time graph (the target graph ). 3D. increased gravitational field of neutron star. 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Note: This simulation was updated (10/25/22). Astudent is conducting an expirement to determine how far a ball will roll down a ramp based on the angle of the incline what is the independent variable and dependent. Uniform Acceleration: Ball Rolling down an Incline -- xmdemo 111 - YouTube Explanation will be at http://xmdemo.wordpress.com/111Catalogue at https://xmphysics.wordpress.comFollow me on. That would take a long time! So recapping, even though the speed of the center of mass of an object . If you dropped a ball from your hand straight down, what would be the acceleration of the ball? To calculate the acceleration of the ball, you can use the equation a = (V 1 - V 2 )/t *. Ramp 'n Roll. Take advantage of the WolframNotebookEmebedder for the recommended user experience. If the ball is rolling without slipping at a constant velocity, the point of contact has no tendency to slip against the surface and therefore, there is no friction. Where do you think it's going? Use this worksheet to give sixth-grade math learners practice finding perimeter on the coordinate plane! Bushra S, Alaris W, Tierra C Mr. Sponagle SPH4U-02 Preformed on September 14, 2022 Due September 19, 2022 Proportionality of a ball rolling down a ramp Purpose: Determining how long it takes for a ball to roll down a ramp when being dependent on the length and steepness of said ramp. Spanish-English dictionary, translator, and learning. 9. This program is supported in part by the National Science Foundation (DMR 21-44256) and by the Department of Physics. Because we know that V = t/x, we can calculate the velocities across each distance x. et dcouvrez des images similaires sur Adobe Stock. Use the Incline Angle slider to adjust the angle of the incline. Forces are vectors and have a direction and a magnitude. $\endgroup$ - please delete me Aug 6, 2013 at 6:27 Wolfram Demonstrations Project There are two limiting cases, one with no friction and one with friction, so there is no slippage of the ball. Published:June32014. Ever wished to ride in lamborghini aventador with an adventure of thrilling drift car crash. Give feedback. Note: in this simulation it is assumed that the coefficient of static friction is sufficiently large to cause rolling without slipping. Powered by SiteManager | Contact Webmaster. Year = {2000} Therefore, only the component of the gravitational force which points along the direction of the ball's motion can accelerate the ball. . Moment of Inertia: Rolling and Sliding Down an Incline This is a simulation of five objects on an inclined plane. This demo can also be used to show the relative static friction coefficients of different materials on wood. With friction, there is both translational and rotational kinetic energy as the ball rolls down the ramp. Ball sliding down a ramp. The AIP Style presented is based on information from the AIP Style Manual. Author = "Naoki Mihara", Copyright 2023 Education.com, Inc, a division of IXL Learning All Rights Reserved. Missing units were added as well as a few other fixes. Use the protractor to measure the angle between the ramp and the floor. This can be seen in Make about a 10 cm height difference between the ends of the ramp. The MLA Style presented is based on information from the MLA FAQ. Base of the ramp. Set the golf ball at a measured distance along the ramp. Disk Sliding or Rolling in a Semicircular Well, Shooting a Ball from a Block Sliding Down a Ramp, "Effect of Friction on Ball Rolling Down a Ramp", http://demonstrations.wolfram.com/EffectOfFrictionOnBallRollingDownARamp/, Dan Curtis (Central Washington University), Alexi Radovinsky, and Stan Wagon (Macalester College), Effect of Friction on Ball Rolling Down a Ramp. A cylinder, sphere and hoop rolling down a ramp. You will need to take eight different time measurements and will calculate four velocities and two accelerations. x is the distance between the marked points. You can then compare the accelerations you calculate to see if the acceleration along the ramp stays constant (which it should). This resource is stored in 2 shared folders. Description The simulation beeps each time the ball passes one of the vertical red lines. Connecting simple harmonic motion and uniform circular motion; A ball on a spring; A ball on a spring - energy graphs; A ball on a spring - with damping (friction) Then send your curated collection to your children, or put together your own custom lesson plan. 20. Plug-ins. Rolling down a ramp Plot energy as a function of The object is a The object rolls without slipping down the ramp. Learn all about dilations on the coordinate plane with the help of this one-page handout! @misc{ Mihara, Naoki. This Demonstration was written in Making Math. 3 cm 77 cm 40. This is because sin() [when it is between the values 0 and (/2)] will increase with an increasing. Use this one-page reference sheet to help students learn all about translations on the coordinate plane! If a ball is running down a ramp, why is it that when you change the height of the ramp, the ball runs down the ramp faster? Method Set up a ramp balanced on a wooden block at one end. In other words: He was the inventor of the telescope, and one of the first people to suggest that the Earth traveled around the Sun and not the other way around. Calculate the acceleration for the points you tested using the equation. You dont want them too long because you want to leave time for the ball to accelerate between whereyou are calculating velocities, so they should be between 10 and 15 cm each. . Because there is a greater force pulling the block down the plane, a steeper incline will cause the block to begin descending when it may not have on a shallower incline. Updated 7-18-2017 (block instead of a ball) by AD This is a simulation of five objects on an inclined plane. The cube slides without friction, the other objects roll without slipping. The applet then displays the motion of the ball as well as position, velocity, and acceleration graphs in real time. Try our coordinate plane worksheet with your kid. The kinetic energy in A is 10 J, in B is 30 J. You may also want to do some test rolls to work the values out - i.e. Uniform Acceleration in One Dimension: Motion Graphs, Position, Velocity, and Acceleration vs. Time Graphs, Kinematics Graphs: Adjust the Acceleration, Kinematics in One Dimension: Two Object System, Projectile Motion: Tranquilize the Monkey, Friction: Pulling a Box on a Horizontal Surface, Static and Kinetic Friction on an Inclined Plane, Inclined Plane with Friction, Two Masses, and a Pulley, Conservation of Mechanical Energy: Mass on a Vertical Spring, Momentum & Energy: Elastic and Inelastic Collisions, Center of Mass: Person on a Floating Raft, Simple Harmonic Motion, Circular Motion, and Transverse Waves, Wave Pulse Interference and Superposition, Wave Pulse Interference and Superposition 2, Wave Pulse Reflection (Free & Fixed Ends), Air Column Resonance with Longitudinal Waves, Electric Circuit with Four Identical Lightbulbs, Equipotentials & Electric Field of Two Charges, Rotation: Rolling Motion Basics + Cycloid, Moment of Inertia: Rolling and Sliding Down an Incline, Rotational Inertia Lab (choice of three scenarios), Equilibrium Problem: Bar with Axis Supported by a Cable, Angular Momentum: Person on Rotating Platform, Fluid Dynamics and the Bernoulli Equation. This demonstration shows constant acceleration under the influence of gravity, reproducing Galileos famous experiment. Number = {3 March 2023}, The applet then displays the motion of the ball as well as position, velocity, and acceleration graphs in real time. It can also be used in rotational dynamics [for a discussion on rotational dynamics, click here],to show and calculate moment of inertia, angular velocity, angular acceleration, and angular momentum. Related. Login to leave a comment sharing your experience. It is important to note here that the angle of the inclined plane will be the same as the angle between the force of gravity and the force perpendicular into the plane. A greater will require a greater force (and therefore a steeper incline) to begin moving than a smaller . 1. Put time on the x-axis, and distance traveled on the y-axis. Galileo Galilei was a physicist, astronomer, mathematician, creative thinking mastermind who lived in the 16th and 17th centuries in Italy. Photos Illustrations Vecteurs Vidos Templates Gratuit Polices. This site provides a simulation of a ball rolling on a segmented ramp. Galileo's hypothesis was that balls rolling down ramps of equal height would reach the same velocity as a free-falling ball no matter the slope (steepness) of the ramps. B. To calculate the acceleration of the ball, you can use the equation a = (V1 V2)/t *. Relevant Equations: Consider the situation in the attached photo. The number of people accessing the page since then is: Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, http://physics.bu.edu/~duffy/classroom.html. Adjust the stack of books until you can get the ramp as close to 30 as possible. %A Naoki Mihara %T Ramp 'n Roll %D 2000 %I Wisconsin Society of Science Teachers %C Oshkosh %Uhttp://www.laboutloud.com/rampnroll/ %O text/html, %0 Electronic Source %A Mihara, Naoki %D 2000 %T Ramp 'n Roll %I Wisconsin Society of Science Teachers %V 2023 %N 3 March 2023 %9 text/html %Uhttp://www.laboutloud.com/rampnroll/. The dynamics of a ball rolling down an incline is interesting. Description Year = {2000} The center of mass is gonna be traveling that fast when it rolls down a ramp that was four meters tall. Bookmark this to easily find it later. The object slides down the ramp. Author = "Naoki Mihara", Explore forces, energy and work as you push household objects up and down a ramp. Time how long it takes for the golf ball to hit the floor after your let the ball go. N. Mihara, Ramp n Roll (Wisconsin Society of Science Teachers, Oshkosh, 2000), . Height of the ramp. Avoid making the ramp too. Graphs show forces, energy and work. The coefficient of static friction () of the block on the ramp will change magnitude of the force (F2) necessary to begin the block sliding. The cube slides without friction, the other objects roll without slipping. Written by Andrew Duffy. 3 cm 77 cm 20. If you increase the steepness of the ramp, then you will increase the The user can set the ball's initial position and velocity and the geometry of the ramp. To do this you will want to mark out eight evenly spaced marks on the ramp and take note of the time that the ball crosses each mark (Image of what the ramp should look like below). People easily intercept a ball rolling down an incline, despite its acceleration varies with the slope in a complex manner. The force of gravity points straight down, but a ball rolling down a ramp doesn't go straight down, it follows the ramp. Powered by WOLFRAM TECHNOLOGIES two different ways: University of Illinois at Urbana-Champaign. We use cookies to provide you with a great experience and to help our website run effectively. Mihara, Naoki. The site also provides drawing tools for users to draw graphs by hand that match the simulated motion. You will not measure this acceleration because of the inclined plane, but if you were to conduct an experiment by dropping balls from different heights, this is what you would expect. Fans should climb this ramp until they reach the walkway that bisects it, using Stasis to . This Demonstration shows the translational velocity of a ball, projected in 2D, as it moves down a ramp. The site also provides drawing tools for users to draw graphs by hand that match the simulated motion. Lower and raise the ramp to see how the angle of inclination affects the parallel forces acting on the file cabinet. 1) Components of forces. So we can easily seen that. Use the check boxes to select one or more objects. Use the ruler or meter stick to mark 10 cm intervals along the ramp, starting at the floor and going upward. This site provides a simulation of a ball rolling on a segmented ramp. Enjoy this SUV driving simulator in amazing impossible off-road, mountain, highway & roadway tracks. Explore forces, energy and work as you push household objects up and down a ramp. Record the final angle in your notebook. Simulation first posted on 6-4-2016. Rolling (without slipping) ball on a moving . It is with this anglethat we measure the component forces, F1, and F2. Learners plot (x, y) coordinates on a plane to locate an emergency situation in this fun math game! The APA Style presented is based on information from APA Style.org: Electronic References. This Demonstration shows the translational velocity of a ball, projected in 2D, as it moves down a ramp. The object rolls without slipping down the ramp. Use the check boxes to select one or more objects. Use the mass and radius sliders to adjust the mass and radius of the object(s). This can be seen in the images below: As seen above, a ramp with a larger (incline angle) will have a greater component force vector pushing it down the ramp (F2), and a smaller component force vector that is pushing it directly into the ramp (F1). If you decide to create an account with us in the future, you will need to enable cookies before doing so. The different mass distributions cause the rolling objects to have different rotational inertia, so they roll down the incline with different accelerations. Because timing and other factories like wind resistance are an issue at great heights (like dropping a ball from the height of a building), Galileo and fellow scientists used inclined planes, like ramps, to conduct their experiments. As players continue through the Owa Daim Shrine, they will encounter a large ball rolling down a ramp. By using this website, you agree to our use of cookies. Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. }. Contact us! Tlchargez la photo Father helping child roll bowling ball down a ramp at bowling alley. In this wrecking Apparently, however, they are poor at detecting anomalies when asked to judge artificial animations of descending motion. Volume = {2023}, ], A greater force acting on the block can be created by increasing the angle () of the ramp. Publisher = {Wisconsin Society of Science Teachers}, This site provides a simulation of a ball rolling on a segmented ramp. This seems like a difficult task! acceleration of a ball which rolls down the ramp. The Science behind a Ramp. N. Mihara, (Wisconsin Society of Science Teachers, Oshkosh, 2000), WWW Document, (. The user can set the ball's initial position and velocity and the geometry of the ramp. Use the mass and radius sliders to adjust the mass and radius of the object (s). The different mass distributions cause the rolling objects to have different rotational inertia, so they roll down the incline with different accelerations. What is the time for the ball to roll from 200 cm to 250 cm? The Chicago Style presented is based on information from Examples of Chicago-Style Documentation. Simulation first posted on 1-4-2017. Help your little one practice shape identification in this worksheet where he'll find and color the different kinds of shapes you might encounter on a plane. A really simple way to solve the dynamics of this system is to split the ramp into, say, 100 elements then compute the acceleration of the ball at the start, integrate the acceleration to get the velocity at the next point. How is the national wildlife refuge system similar to the pacific region coastal program? Galileo and many of his contemporaries are thought to have begun experimenting with falling objects and testing the idea that even though objects have different masses, they will fall towards the Earth at the same velocity. You can then compare the accelerations you calculate to see if the acceleration along the ramp stays constant (which it should). As F2 increases with increasing , it will allow blocks with greater coefficients of static friction to begin to slide down. Wolfram Demonstrations Project & Contributors | Terms of Use | Privacy Policy | RSS Simulation first posted on 1-4-2017. Introduce your child to the inclined plane, one of the six simple machines that helps to make work easier for us! And similarly for t3 and t4. Mark out 30 cm at the end of the ramp. When there is no slippage, the ball slides down the ramp with no rotation. by I am posting my animations on this channels for people to see and critique. The acceleration at each point should be almost the same. 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