The independance of horizontal and vertical motion

[Cheman]

The independence of horizontal and vertical motion...

Obviously it is possible to prove the independence of horizontal and vertical motion empirically - we only need look at projectile motion following a parabolic path. However, I have never found a convincing algebraic proof the for the independence of these 2 types of motion.

I have asked my physics teacher and he says that it can apparetly be proved by treating Gravity as a Vector, and then assessing the overall motion of a body - apparently things like "gcos90" appears, which obviously equal "0", and this can be used to show the independence of horizontal and vertical motion.

If anyone could supply me with a convincing algebriac proof i would be really greatful!

Thanks in advance.

 

[ZapperZ]

Obviously it is possible to prove the independence of horizontal and vertical motion empirically - we only need look at projectile motion following a parabolic path. However, I have never found a convincing algebraic proof the for the independence of these 2 types of motion.

I have asked my physics teacher and he says that it can apparetly be proved by treating Gravity as a Vector, and then assessing the overall motion of a body - apparently things like "gcos90" appears, which obviously equal "0", and this can be used to show the independence of horizontal and vertical motion.

If anyone could supply me with a convincing algebriac proof i would be really greatful!

Thanks in advance.

What do you mean by "algebraic proof"? If you accept the mathematical concept of a vector, then if you have the vector oriented along the x-axis, can you find the component of the vector along the y-axis?

Zz.

 

[Claude Bile]

The x and y unit vectors are orthogonal, thus the x and y components can be treated separately. A ZapperZ pointed out, there really is no algebraic proof to consider.

Claude.

 

[pervect]

You also need to define how vectors multiply before you can say that$$\hat{x}$$ and $$\hat{y}$$ are orthagonal. A metric such as

ds^2 = dx^2 + dy^2

is one way of giving the necessary definition of the vector product $$\hat{x} \cdot \hat{y}$$, and a diagonal metric such as the specific example above is necessary and sufficient to make these two vectors orthagonal.

 

[HallsofIvy]

You can't prove a physical fact mathematically! They can only be proven by experiment (what you called "empirically").

(If you assume that physical velocity can be represented by mathematical vectors, then you can use the properties of vectors. Of course, you would have to base that "assumption" on experiment.)