Pythagoras and physics 1

Pythagoras appears in many places, the objective here is more than "prove" Pythagoras is to see how it connects to familiar concepts in physics, and what physics has to say about it! For more classical approaches look in this post.

Hypotenuse is "a", leg "b", leg "c", so Pythagoras means that:

a ² = b ² + c ²

Lets try to use physics to show Pythagoras. Since we have something squared we can look for something that is squared in physics...

Energy = 1/2m*v²

Remember that the velocity is a vector (we will use bold letters for vectors) and the formula for energy just uses the length or absolute value of the vector. We will choose m=2 then Energy = v ² to make calculations easier.

Over the sides of the triangle lets put the vectors a, b, and c. Notice that a=b+c!

If E(a) means the energy of an object with velocity "a" so Pythagoras says that:

E(a)=E(b)+E(c)

Why should that be true? Remember that "b" and "c" make 90 ^o . If they don't then it is actually not true!! Because we should replace Pythagoras by the cosine rule.

So why it should be true when "b" and "c" are perpendicular?

The energy of an object is also the work needed (that is force * distance) to make it go from rest to that velocity.

Lets apply a force horizontally to make an object acquire velocity b then its energy is E(b)= b ² . Now lets apply a force vertically to make it get an it acquire an upward velocity c. Contributing with an energy E(c)= c ² adding the two vectors "b" and "c" we have the vector "a" that we know should have energy E(a)=a ² .

Voila!! E(a)=E(b)+E(c)

Why does this argument works? Notice that the second time we apply the force (now vertically) that force doesn't change velocity b horizontally so all it work (energy) is in the upward direction.

Or more precisely:

Forces perpendicular to the movement don't do any work because they don't change the absolute value of the velocity. But they do change the velocity's direction! Recall the circular movement.