# The independance of horizontal and vertical motion

1. ### Cheman

236
The independance of horizontal and vertical motion....

Obviously it is possible to prove the independance 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 independance of horizontal and vertical motion.

If anyone could supply me with a convincing algebriac proof i would be really greatful! :tongue2:

2. ### ZapperZ

30,004
Staff Emeritus
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.

3. ### Claude Bile

1,479
The x and y unit vectors are orthogonal, thus the x and y components can be treated seperately. A ZapperZ pointed out, there really is no algebraic proof to consider.

Claude.

4. ### pervect

7,947
Staff Emeritus
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 defintion 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.

5. ### HallsofIvy

40,549
Staff Emeritus
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.)