Article status: Draft
Time Estimate for Reading: 30 min
Learning Objectives: Importance of Acceleration in understanding physics
Effort Required: Medium
Pedagogy Model: Evolution, Formula Analysis, Interdisciplinary
Prior Physics Concepts: displacement, velocity, acceleration, mass
Prior Math Tools: Secondary school level Arithmetic, geometry and algebra
Every student who continued beyond primary school would know this story. The story of Newton sitting under a tree, observing an apple fall and coming up with the theory of gravitation. We chose to remember the story and may be did not give enough attention to the concept behind this story. Even we went on to sending a little speck of the tree to Space (a part the same apple tree has been taken to space in 2010 )
The concept behind the falling apple is acceleration and Newton (1642 - 1727) would have got the idea of using acceleration as the core concept, to bridge space, time and mass; eventually leading to the three laws of motion (dynamics) to analyze and predict motion on earth (terrestrial mechanics) and Planetary dynamics.
Take at look at the seemingly simple mathematical relationships between space, time and mass.
Displacement = acceleration * time^2Uniform Velocity = Displacement/time
uniform acceleration = Change in velocity/time
mass = force/acceleration (m = f/a)
We should also make note of sinusoidal acceleration and Jerk. Jerk is abrupt change in acceleration (The change is neither continuous nor sinusoidal)
Apart from being a bridge, acceleration ss significant because, nature took close to 2000 years to reveal this story.
To understand acceleration, we need to begin with Euclidean point (geometry- 350 BC), The Galilean Falling ball experiment (1590), The Galilean Telescope (1620), The Kepler point - (1571 to 1630) and for the sake of completeness; the Planck Point - 1900 AD (the Measure of Minimum Possible Energy). With the introduction of Planck Point, there are no infinitesimals anymore. A Point is no more abstract and seems to have a definite, discrete value of Planck Point.
Let us time travel to 300 BC. Euclid has given a formal account of geometry (space) and it was made possible by the abstract concept of point. Abstract in the sense that something that is not natural, that cannot be physically experienced, that which is not intuitive or that which is beyond common sense. Now science gets beyond common sense.
Many years pass by. 2000 years. Quite a long time and we reach the times of Galileo. Galileo establishes the relationship between space and time with his vertical falling ball experiment (acceleration), the horizontal motion (rest and constant velocity motion does not require an external force), a combination of vertical and horizontal motion leading to parabolic motion (all of these are formalized using geometric mathematics and thanks to Euclid).
Galileo, with his experiments, more or less, helps define the concept of acceleration (and of course velocity) It is unbelievable that just linking up space and time took such a long time (and from there it took hardly 30 years for Newton to define mass and link up space, time and mass).
Why did it take so much time?
Let us begin our journey by first getting to terms with distance, displacement, velocity and acceleration.
To begin with, Descartes (1637) helps us with the concept of coordinate geometry. The euclidean non-existent point can now can be defined with well established co-ordinates (x, y, z) with respect to a fixed origin.
With the definition of Descartes "co-ordinate" point, Distance can be measured between two points. The measurement being made in stadia, ft or the SI units, The distance may be measured across any path. Notice that we are using the word path instead of straight line.
Now let us define and we need to introduce one more word "displacement". Displacement is the shortest distance between any two points. In real life, there cannot be any displacement without considering the time interval (ah. that's velocity). Hence displacement is again an abstract concept.
Having defined displacement, Velocity can be defined as the displacement (measured in meters) in unit time (measured in seconds). Velocity is given the unit of meters/second. Simple. Notice here that, space (m) occupies the place of numerator (may be because, those days, we were yet to get to terms with time.)
Having defined velocity, Acceleration can be defined as the rate of change of velocity. Now there are two choices; rate of change with respect to space or with respect to time. This was the fundamental problem in the past; the past before the days of Galileo. Those were the days when relationship between space and time were yet to be established.
Choice 1: Let us try to define acceleration as rate of change with respect to space. This is what we get. acceleration = velocity/space = (meters/second)/meters = m/(ms) = 1/s. Okay. Now, What is the inverse of time?. It is frequency; measured in Hertz. But frequency as a concept was established for analyzing oscillatory motion; and analysis of oscillatory motion had to wait for discovery of differential calculus. Nature chose to reveal the secrets of calculus to newton and Leibniz (and they are yet to be born).
Choice 2: The choice now is to use the rate of change with respect to time. Galileo experimentally proved that, for falling objects closer to the surface of earth, displacement varies with respect to square of time. d is proportional to t^2 (independent of the mass of objects).
Getting back to acceleration, how can the unit of displacement (m) be equated to square of time(s^2). Well, we may replace the proportionality constant with something that would give us just 'm'; a measure of displacement. So, this is the clue to acceleration. Acceleration can be given the units of m/s^2. On a similar note we can also apply this logic to velocity and define acceleration as the rate of change of velocity with respect to time (m/s^2). Fits Nicely.
So far, we have put the concepts of displacement, velocity and acceleration into perspective. Now the challenge is to measure them.
- Displacement (rather length) can be measured with a measuring tape. Simple.
- Velocity can be measured with a tape (length between two points) and a stop clock (end time - start time ). Simple again.
- To measure acceleration, we need final and initial velocities between two points (say point A and Point B). Sounds interesting. It is no more simple. How do we do it? Somehow we need to arrive at a method to measure the velocity at every point. Feeling dizzy.
The way forward is to stretch point A into A1-A2 and Point B to B1-B2 to some known length; the smallest possible length, depending upon the resolution of our measuring equipment. Remember, we should also measure the time interval and the resolution of clock should also be taken into account. With this in place, we can calculate the initial and final velocities. Once we know the velocities and time interval, we can calculate acceleration.
Well. We may now understand (hopefully) and we may know how to measure displacement, velocity and acceleration. With that we can understand the significance of acceleration as the bridge between space and time. A bridge that has taken almost 2000 years to build.
Newton is set out to establish the bridge between space, time and mass.
From Galileo's falling ball experiment, it is fascinating to see that, for bodies of independent mass to fall all the same rate (constant acceleration), earth has to some how know the mass of each object and try to pull them with an appropriate force.
In order to validate this idea, Newton must have come up with an experiment to observe the behavior of mass on a horizontal plane and applying a constant force; the force being exerted by a spring (instead of gravity). This leads to an interesting and expected observation that, acceleration of the object varied according to mass of the objects.
By calibrating a spring with a known reference weight and conducting the horizontal block experiment and measuring the acceleration, mass can be determined.
An excellent account of this experiment can be found in this video by Prof Shankar Ramamurthy of Yale University (it is a 1 hr video. This experiment is discussed between 20min and 40min)
There comes the equation m = Force exerted by the spring/acceleration (F/a). it is important to not here that, force from the spring will vary with displacement. since we are measuring the acceleration at the same point for any mass and also the forces cancel out from the equation, this experiment is considered valid.
We could have also used a hanging mass instead of spring. but we will consider that later (till we cover free body diagrams and interactions)
The definition of mass (inertial or gravitational mass) in this way settles the long standing dispute of mass, density (mass/volume), weight (mass * acceleration due to gravity of 9.81 m/s^2) and force (1 kg mass accelerated at 1 m/s^2 acceleration).
By rearranging the formula, we can also arrive at f=ma. The formula that is most misunderstood.
With such a significance, we may say that as much A for apple is for babies, A for acceleration should be introduced to beginners of science.
Foot Note.
With this in place we may move to understand the concepts of instantaneous velocity and acceleration
Instantaneous velocity = infinitesimal displacement over infinitesimal time interval.
Instantaneous acceleration = infinitesimal change in velocity over infinitesimal time interval.
Now, the interesting question is, how small is the smallest possible length?. We cant just say "depending on the resolution of measuring equipment".
Mr Kepler provides a clue. The Infinitesimal. The smallest possible length which is very close to zero and not exactly zero. For this we need to begin with Archimedes (350 BC). This will be done in another article. Kepler is mostly known for establishing the laws of planetary motion. But he also established the more important concept of infinitesimal point (Archimedes in 300 BC is known to have advanced the concept if infinitesimals.). The infinitesimal point forms the basis for non intuitive concepts of instantaneous velocity, instantaneous acceleration and also to the concept of calculus (calculus deals with arithmetic with infinitesimal quantities).
Who said science is common sense? We are dealing with abstract points and infinitesimal lengths. Add to it the concept of Planck Point. A Planck point being the volume of a cube made of Planck Length (not sure of the reason for not using a sphere to calculate the Planck volume. for us a sphere is a more natural shape than a cube. May be nature uses cube as the building block at that microscopic level. Planck length is in the order of 10 raised to the power -35). May be it is time to redefine the concept of infinitesimals, instantaneous velocity and instantaneous acceleration.
As though, this is not enough, Einstein comes up the concept of curved space-time. Space and time are not independent anymore.
Apart from this, the story of mass does not end here. Once we understood that matter is made of atoms and molecules, we redefine chemical mass as "amount of matter" (we have to wait for Avogadro to relate gravitational and chemical mass). We also have Einstein to come up and say mass and energy are equivalent. E=mc^2, the most bizarre equation. Energy is another abstract concept.
So we have abstract concepts like point, displacement and energy, the concepts that are not real. We cannot feel them. But we need to use them to deal with nature.
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