Last week, I showed how one could derive 3 of Newton’s equations of motion. As a colleague of mine pointed out to me on FaceBook, the 3rd equation I showed can, in fact, be derived using algebra from the 1st and 2nd. Also, if you go to the Wikipedia page on the equations of motion, you will find 5 listed, not just the 3 I showed. It turns out that the only two “fundamental” ones are the 1st and 2nd that I showed, the other three (including my 3rd, shown as the 4th in the image below) can be derived from the 1st and 2nd ones.
This is often the case in mathematics, there is more than one way to do something. So, although the method I showed to derive the 3rd equation is perfectly correct, and it also shows how we can rewrite , which is a useful technique, I will today show how it can also be derived by algebraically combining equations (1) and (2).
Remember, equations (1) and (2) were
The first step is to square equation (1), which gives us
Next, we multiply equation (2) by 2 to get rid of the fraction
We next multiply each term in equation (3b) by to give
Comparing equations (3a) and (3c) we can see that we can substitute the 2nd and 3rd terms of equation (3a) by and so we have
which is our equation (3) from the previous blog. As they say in mathematics, Quod erat domonstradum (QED) 🙂