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Calculus in Physics: Newtonian physics & Special Relativity
In this article, we will have a taste of how physicists utilize the power of mathematics through the lens of Taylor polynomials.
We use Taylor polynomials as an approximation to functions to derive the formula for kinetic energy as most people know it.
According to Einstein’s theory of relativity, the mass of an object moving with velocity v is related by the rest mass m0, and the speed of light c.
The kinetic energy of the object is the defined as the difference between its total energy and its energy at rest.
Our objective is to show that when v is very small compared with c, this expression for K agrees with classical Newtonian physics:
How should we do this? Let’s start with the expression given for K and m.
We now factor out m0c².
We rewrite the square root as a negative power so that we can consider the Maclaurin series.
x = -v²/c²
(1 + x)^(-1/2)
Let’s first compute the Maclaurin series in terms of x.
We can now substitute x = -v²/c² into the above expansion.
Let’s consider what happens when v is much smaller than c, which it usually is.
When v is significantly less than c, all terms after the first are very small when compared with the first term. If we omit those later terms, we get a very familiar looking formula!
And that’s our answer.
How amazing 🐉
What was your thought process this time? Comment down below, I am eager to know :) 🏹
