Introduction: THE KINEMETER: Part 1: the Kinematic Tool
HELLO!!! MAKERS,VIEWERS & FOLLOWERS ALIKE! Behold my newest and biggest project: THE...KINEMETER!!! It measures velocity, it measures acceleration, and best of all, it is completely DIY!!!
This is a three part series:
Part 1: You are here. And this is about the physics behind the equipment
Part 2: Teaches the assembly of laser-cut parts. See here
Part 3: Teaches Hardware & Software of the equipment. See here
Before I go into any detail, a brief reflection on the history of kinematics.
How did Galileo & Huygens prove that objects fall towards Earth at 9.8 m/s/s? Those who are history fans know that they used inclined planes and pendulums to get a precise answer for g (acceleration of gravity).
A classic and perfect example of raw derivation, but, can we directly observe an object's acceleration? To answer that question, I worked for a year on a brand new device for directly measuring acceleration.
Most of the project's inspiration came from a class setup I had in school where two sensors were placed on two separate ends of a 1.5 meter long aluminum ramp. The precision of measuring acceleration is superb. Excellent. But, like every piece of technology, the system had its limits.
Due to it's size, one could not modify the setup for any particular experiment, limiting its data capturing capabilities. After looking at the system's individual parts and prices, I realized that our school did not pay a light amount to give its physics students a good education. This made me think of other schools that did not have the luxury of such equipment, let alone an understanding of physical laws. I thought: "shouldn't all schools have access to accurate, versatile tools that could help in teaching students?"
I wanted to make a tool that could revolutionize physics experimentation. And I made one, like I do. Watch the video of my introduction to the brand new Kinemeter and see its potential for yourself.
Hoped you enjoyed that video demonstration. But, if you are a faithful viewer and a maker down to the core, then you probably are wondering: A) What laws of physics does the machine depend on? B) Where do I get the plastic and how do I assemble the pieces for the body? C) How do I build the circuitry to get the machine to work?
I will be answering the first of these three questions today in the first of a three part marathon of creativity. WELCOME!!! To the KINEMETER TRILOGY!!!!!!
Step 1: The Key Laws of Physics Used
In order to build the Kinemeter (either the large or small) I had to base my logic on phenomena that actually happens in nature that can be supported by raw data.
The first law is that if an object is placed on an inclined plane or in free fall, assuming little friction, then the object will accelerate at a constant rate. While this seems a bit empirical, I urge you to test this out for yourself. For those who are aware of this phenomena, great! I illustrated this on the first diagram.
Acceleration: the rate at which an object's velocity changes. If it is 0 m/s/s, then the object is either at rest or is traveling at a constant velocity. If the acceleration is (for example) 1 m/s/s, then for every second, an object travels 1 m/s faster.
The second crucial law is that with constant acceleration, the average velocityis an instantaneous velocity of the object at (T initial+Tfinal)/2 . This can be derived using basic math and common sense using the the second diagram above.
Average velocity: The total displacement of space divided by the delta time. On a V-t graph, this can be also represented by taking the (Velocity initial+Velocity final)/2.
Instantaneous Velocity: On a V-t graph, this is any particular velocity that an object was traveling at in a moment (zero delta time). On an x-t graph, instantaneous velocities can be represented by finding the tangent line to the object's path.
IF YOU ARE STILL CONFUSED, PLEASE SAY SO IN THE COMMENTS!!!
If enough of you want deeper explanations, then maybe I will release a separate instructable going into further detail on these physical laws.
Let's see how I applied these assumptions for the Kinemeter.
Step 2: Kinemeter Disection
Before I explain how it can accurately measure acceleration or velocity, lets see what makes it tic, so to speak. Each letter on the image is explained below:
A) Every sensor is separated from one another by a fixed amount. It is easy, for me personally, to separate them to be equally spaced.
B) Between a sensor and its corresponding transmitter, there will be an infra-red light beam. This is the mechanism that will be used to detect whether an object passed it or not. If the light is received, then nothing happens. If an object passes through the beam, blocking the beam, the sensor that is broken will record the time (at t= Y sec.)that the object broke the beam.
C) The object passing through the sensors (either accelerating or traveling at constant velocity)
D) A broken IR beam. When a beam is broken, the sensors record the time at which the beam was broken.
E) These are your sensors that detect the IR beam. Across from them is a corresponding transmitter. These transmitters give off the IR light beam. The number of sensors you want is up to you, so long as it is at least 3. Two sensors will not give you enough data to calculate acceleration.
Step 3: Now Here's the Scenario:
Now let's say that object is sent through the Kinemeter and breaks the three sensors from the earlier diagram. The object breaking the sensors hit each one at t1, t2, t3. Each sensor is x cm apart.
To generalize this proof to accelerating and non accelerating objects, I will only use these variables. Remember, this machine is mostly effective for determining constant accelerations, so we will assume that is the case.
The arduino that runs the Kinemeter knows the distances between each sensor and knows the times when the object passed each sensor.
Now how can a simple machine use only this data to calculate acceleration?
LET'S FIND OUT!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Step 4: Let's See the Math
In the code, the arduino takes the times it is given, and finds the time it took for the object to travel between sensors. That gives it the following:
Delta t1 = t2-t1 Delta t2= t3-t2
Since it has two delta times, and the arduino has the two displacements (x cm), the arduino can calculate the average velocities for those two time periods. That gives the following:
Avg. Velocity1 = x cm/(t2-t1) s Avg. Velocity2 = x cm/(t3-t2) s
From what I described earlier, average velocity is the instantaneous velocity of an object at the average of two times. Since this is the case, the arduino then knows that:
The object was travelling at Avg. Velocity 1 at T1= (t1+t2)/2and was travelling at Avg. Velocity 2 at T2=(t2+t3)/2
This bizarre method of averaging the times to make average velocities instantaneous velocities is the engine of the Kinemeter.
AND NOW THE PROCESS TO DETERMINE ACCELERATION IS...............:
Average & constant acceleration can be taken by dividing the change in Velocity by the change in time. The final step the arduino performs to calculate the acceleration of an object is the following:
Acelleration = ((Avg. Velocity2) - (Avg. Velocity 1))/(T2-T1)
Those who want to see the full expansion and simplification would end up with this:
Acceleration = ((X/(t3-t2)) - (X/(t2-t1)))/ ((t3-t1)/2)
And what do you know, you solved for acceleration. Just plug in values in their respective places.
See the diagram for a visual representation of this experiment.
Step 5: WHY NOT D= (1/2)at^2 + Vt??
One equation you might be familiar with is D= (1/2)at^2 + Vt, where D is displacement & V is the initial velocity.
This equation seems far simpler than the monstrosity of an equation I gave earlier. SO...Why not use this above equation?
The answer is simple. To use this equation, one would need to be perfect in dropping or rolling an object. Why? because of that new variable: Vt. We have no clue what Vt is. Since in an experiment, we want to reduce the unknowns to just one, the acceleration, then guess what, you would have to drop the object directly above the first sensor, no space, no mistake, or else the initial velocitywill be different each time, resulting in different acceleration values. NOT GOOD!!!!!!!!
Sure, the previous equation might not look pretty, but think of it this way: it will give you consistent results for a particular situation. You do not have to be perfect in setting up an experiment. In sacrificing the simplicity of the math behind the Kinemeter, you are also losing the burden of keeping the experiment perfect while having the Kinemeter accurately calculate the motion of an object for you.
Step 6: On a Final Note....
The potential for such a device is uncalculable. This device is not only a simplification, but a solidification of the laws of physics. This device is also a robust system (as seen in the video demonstration). The equation I derived for you allows users of this remarkable machine to have flexibility in how one performs an experiment.
Messed up with dropping an object? That's OK! The Kinemeter will most likely still give you promising results. But do not mistake this robustness for sloppiness, the Kinemeter still can measure velocity and acceleration with high precision.
A device like this, though, does not belong in the hands of one man, heck, even one company. It belongs to the hands of students who don't have easy access to this kind of educational opportunity. What I am trying to say is that as a teacher and maker, it is my obligation to teach you, the viewers and followers, how to build this machine, as well as code it.
STAY TUNED, FOLKS!!! Because next time, I am going to show you how to actually build your very own Kinemeter, and how you, the people, have the capabilities of remodling the origional design for your own benefits.
Thank you for watching and supporting me in 2015.
SEE YOU NEXT TIME!!!!!!!!!!!!!!!!!!!!!!! AND HAPPY NEW YEARS!!