That we can take derivatives to shift from position functions, to velocity functions, and then to acceleration functions. That the entire
world is governed by derivatives. That we can figure out why some math is actually important to the real world. It exists to torment
students and lower GPAs.
We discuss one of the original uses for calculus, how position, velocity, and acceleration are related.
Video Lecture
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We’ve discussed how first and second derivatives can be envisioned geometrically, and how to find them analytically. There are virtually unlimited areas where these ideas can, and routinely are, used - which is why calculus is considered important enough to be part of the standard mathematical literacy progression for students. But one of the most enlightening examples is also one of the original examples - the relationship between position, velocity, and acceleration.
When Sir Isaac Newton first developed his ideas of calculus, his goal was to develop a mathematical model of gravity. In particular, he wanted to create mathematical tools that would allow him to represent movement of an object as it was subjected to gravity. The resulting model of physical reality came to be called Newtonian mechanics - the basic physics that largely model the macroscopic world, i.e. the world of “large” objects (stuff that isn’t microscopic).
His vision, which we now have the mathematical tools to understand, revolved around the following idea: The rate of change of position is velocity, and the rate of change of velocity is acceleration. Which means that, if we can model the position at time of an object, we can take a derivative of that function to immediately get a function for the velocity at any time of that object, and moreover we can take a second derivative to immediately get a function for the acceleration at any time of that object.
And so the relationship was (mostly) cemented. Using the conventional notation we have the following:
Newtonian Position/Velocity/Acceleration Let be a function representing the position of an object at time in seconds.
Then:
- is a function representing the velocity of that object at time in seconds.
- is a function representing the acceleration of that object at time in seconds.
1 : A core element of Newtonian Mechanics is...
There will be several example videos detailing this relationship and how to move from one function to the other, but for now it is helpful to see these things visually. This is how mathematicians of the time (including Newton) would start the process of developing this area of math - by visualizing what is happening and trying to ascribe mathematical ideas to the real-world events. Watch the above video for a nice visualization of this process!
We have covered one of the most fundamental uses of calculus, and specifically the derivative - the introduction of Newtonian Mechanics. Indeed, calculus as we know it was largely invented to understand this specific relationship, although it turned out to have applications far beyond even Newton’s wildest dreams. This example helps build useful intuition as to the relationship between a function, its first, and its second derivative.