Prove Perpendicular Components of Motion are Independent

[Tsunoyukami]

Consider a simple textbook problem in two dimensional kinematics - say, a projectile motion problem. I know that the x- and y- components of motion are independent of one another but I don't understand why. I know this is true due to everyday observation - empirical evidence of this being the way things are - but is there a way to prove this to someone who might be skeptical?

 

[Bandersnatch]

These might help:
http://en.wikipedia.org/wiki/Linear_independence
http://en.wikipedia.org/wiki/Basis_(linear_algebra)
(remember that position, velocity, acceleration are all vectors in 3D space)

If you hadn't taken linear algebra yet, it's basically saying that a space of N dimensions is defined by N vectors, such that neither of which can be represented by a combination of the rest, and all other vectors can be represented as the combination of these N vectors.

Or, in other words, that (on a flat plane) you can't get anywhere East nor West by going North or South.

 

[Tsunoyukami]

Thanks! I'm familiar with these concepts and I'm not sure why I didn't think of them in the first place!

Is there another simple explanation one could offer when explaining this idea to students unfamiliar with linear algebra? I've been asked about this concept by several friends taking high school level physics and, though the notions of linear independence and basis are not overtly difficult, they do require some knowledge of linear algebra that goes what might be called "significantly" beyond the high school curriculum. (For example, I personally learned of the independence of x- and y- components in Grade 11 (10, maybe?) and learned about linear independence and basis in my first real algebra class two (three?) years later while at university.)

Thanks again!

 

[Bandersnatch]

Is there another simple explanation one could offer when explaining this idea to students unfamiliar with linear algebra?

I'm trying to think of something other than the E-W/N-S example. Maybe ask them how long it would take for a rock thrown up to fall down (or how high it'd get) if it were thrown vertically (or dropped) by a stationary person versus by a person in a speeding train?
You'd need to make sure they don't get confused between the two reference frames.

I'll think about it some more.

or: when shooting a gun (horizontally level barrel) - does high muzzle velocity make the time for the bullet to fall longer than if it were just dropped? You can segue into what determines range of a gun.

 

[Quantum Defect]

Thanks! I'm familiar with these concepts and I'm not sure why I didn't think of them in the first place!

Is there another simple explanation one could offer when explaining this idea to students unfamiliar with linear algebra? I've been asked about this concept by several friends taking high school level physics and, though the notions of linear independence and basis are not overtly difficult, they do require some knowledge of linear algebra that goes what might be called "significantly" beyond the high school curriculum. (For example, I personally learned of the independence of x- and y- components in Grade 11 (10, maybe?) and learned about linear independence and basis in my first real algebra class two (three?) years later while at university.)

Thanks again!

If, you can write the equations of motion so that the x-terms (v_x, a_x, x) only "talk" to each other, and the same goes for the y and z terms, then there is no way for what is happening with y to effect x or z.

I.e.
F_x = k_x * x
F_y = k_y * y - g * v_y
F_z = 0

Th force in the y direction only changes what is going on in the y direction, and vice versa for the x direction. To understand the motion in the x direction, you need to only solve the first equation. To solve for the motion in the y direction, you need only solve the second equation.

If, on the other hand, you had a situation where the forces look like:

F_x = k*x - g* v_y
F_y = n*y

You now have a "mixing" term in the first equation, where what is going on in y effects what is happening in x. I.e. you cannot say what will be going on with x, until you know what is going on with y -- the equations are coupled.

This is a common occurrence (coupled differential equations) in physics. You will see this a lot in quantum mechanics (time-dependent perturbation theory, particle scattering), except in qm, the coupled equations are not "equations of motion".

 

[Tsunoyukami]

Thanks - I'm familiar with such motion. Of course it's possible to construct more complex problems with more complex (interesting) solutions (motion), but I'm more interested in trying to explain the basic fundamental notion that what happens in each cardinal direction is independent of motion in the other directions to students with little knowledge of physics.

I do like what you've said here:

If, you can write the equations of motion so that the x-terms (v_x, a_x, x) only "talk" to each other, and the same goes for the y and z terms, then there is no way for what is happening with y to effect x or z.

This seems a reasonable way to get students to understand that the motion in one direction is independent of motion in the others. My main difficulty would be explaining why this is true to a skeptical student, without relying upon more sophisticated ideas (linear independence and basis) that they might not be familiar with.

 

[Tsunoyukami]

I'm trying to think of something other than the E-W/N-S example. Maybe ask them how long it would take for a rock thrown up to fall down (or how high it'd get) if it were thrown vertically (or dropped) by a stationary person versus by a person in a speeding train?
You'd need to make sure they don't get confused between the two reference frames.

I'll think about it some more.

or: when shooting a gun (horizontally level barrel) - does high muzzle velocity make the time for the bullet to fall longer than if it were just dropped? You can segue into what determines range of a gun.

Thanks again, Bandersnatch. Those suggestions are pretty typical of what I've come across. I don't have anyone asking me to explain this in detail right now, but I was just curious if there were any simple explanations (proofs) one could offer to students who were skeptical of the claim that the motion really is independent. By having the student think about various situations it can build intuition and feeling, but they might not understand why the motion is independent and just accept it as being true because the teacher said so (which isn't really ideal, in my opinion).

 

[nasu]

I suppose this is a result of the space being isotropic. In anisotropic media the components may be coupled.
Like applying an electric field along x-axis and getting a current density with component along the y axis.

 

[Quantum Defect]

Thanks - I'm familiar with such motion. Of course it's possible to construct more complex problems with more complex (interesting) solutions (motion), but I'm more interested in trying to explain the basic fundamental notion that what happens in each cardinal direction is independent of motion in the other directions to students with little knowledge of physics.

I do like what you've said here:



This seems a reasonable way to get students to understand that the motion in one direction is independent of motion in the others. My main difficulty would be explaining why this is true to a skeptical student, without relying upon more sophisticated ideas (linear independence and basis) that they might not be familiar with.

You might try turning the tables on the skeptical student. Ask her/him if s/he can come up with an argument (using the language of mathematics) for a simple kinematics situation where the motions in the x and the y direction are not independent of each other.

 

[A.T.]

Consider a simple textbook problem in two dimensional kinematics - say, a projectile motion problem. I know that the x- and y- components of motion are independent of one another but I don't understand why. I know this is true due to everyday observation - empirical evidence of this being the way things are - but is there a way to prove this to someone who might be skeptical?

Are you asking about physics or math here? Mathematically they are independent per definition. Physically this model is just an approximation, which fails when for example drag and lift aren't negligible.

 

[Tsunoyukami]

You might try turning the tables on the skeptical student. Ask her/him if s/he can come up with an argument (using the language of mathematics) for a simple kinematics situation where the motions in the x and the y direction are not independent of each other.

That's a great idea, Quantum Defect! Thanks for the input.

Are you asking about physics or math here? Mathematically they are independent per definition. Physically this model is just an approximation, which fails when for example drag and lift aren't negligible.

I was asking the question with respect to physics. I'm aware that we can concoct situations in which the x- and y- components are not independent of one another (ie. coupled equations). The mathematical definition has been suggested above, by Bandersnatch in post #2.

Can you explain what you mean by "Physically this model is just an approximation, which fails when for example drag and lift aren't negligible."? Correct me if I'm wrong, or if I'm misunderstanding you, but even when lift and drag are considered you can break these forces down into their x- and y- components and find the x- and y- components of acceleration which would mean that the x- and y- motion are still independent of one another.
I guess another way to ask the question is:

How can we know for sure that the motion of an object in the x-direction does not affect the motion of an object in the y-direction and vice versa? (This question pertains to a simple physics problems - basic textbook style problems a student would be asked to solve in and Grade 11, 12 Physics or Physics I in a first-year university course. The answer would have to be "simple" enough for such a student to understand, but precise enough that it isn't really beating around the bush and results in an improper understanding.)

 

[A.T.]

you can break these forces down into their x- and y- components and find the x- and y- components of acceleration

You can also decompose vectors into components which aren't linearly independent. But if you choose to decompose them into linearly independent components, then they linearly are independent because you chose to decompose them that way.