Transforming differentials and motion

[fogvajarash]

Hello, I currently have a problem with interpreting how this statement was interpreted:

We have a rate of change which is dv/dt, and in the given notes, they transformed the expression dv/dt into dv/dx dx/dt (using the chain rule). Then, the whole expression simply turns into v dv/dx (as dx/dt is the rate of change in distance which is velocity).

The thing I don't understand is how you can change dv/dt into dv/dx dx/dt. I've noticed that the differentials actually cancel out, but I need the mathematical explanation for this (someone told me that it was due to the chain rule).

Could someone please explain this problem carefully to me? Thank you very much for your patience.

 

[tiny-tim]

hello fogvajarash!

The thing I don't understand is how you can change dv/dt into dv/dx dx/dt. … I need the mathematical explanation for this (someone told me that it was due to the chain rule).

v is a function of x: v = v(x)

x is a function of t: x = x(t)

so we can write v = v(x(t))

now apply the definition of derivative, lim_h->0 [v(x(t+ h)) - v(x(t))]/h, and you get … ?

 

[fogvajarash]

I'm sorry tiny-tim for not replying (midterms are killing me)! The main problem is, how do we know that v is a function of x? (or that is stated in the problem?)

 

[tiny-tim]

hi fogvajarash!

… how do we know that v is a function of x? (or that is stated in the problem?)

in a trajectory, everything is a function of everything else

(you need to get used to this!)

 

[blue_raver22]

I can understand your confusion with this transformation. Let me try to explain it in a way that might make more sense.

First, it's important to understand that the notation dv/dt represents the rate of change of velocity with respect to time. In other words, it tells us how quickly the velocity is changing over a certain amount of time.

Now, when we use the chain rule in calculus, we are essentially breaking down a complex function into smaller, simpler parts. In this case, we are breaking down dv/dt into two parts: dv/dx and dx/dt.

To understand why this is possible, let's think about what each part represents. dv/dx tells us how much the velocity changes for a given change in position, while dx/dt tells us how much the position changes for a given change in time. So, when we multiply these two together, we are essentially finding the rate of change of velocity with respect to position (dv/dx) and then multiplying it by the rate of change of position with respect to time (dx/dt). In other words, we are finding the rate of change of velocity over a certain amount of position and then multiplying it by the rate of change of position over that same amount of time.

Now, you may be wondering why this is useful. Well, by using the chain rule and breaking down dv/dt into dv/dx and dx/dt, we can then rearrange the expression to be v dv/dx. This may seem like a simple change, but it allows us to integrate with respect to position (instead of time) which can be helpful in certain situations.

I hope this explanation helps to clear up any confusion you may have had. The chain rule is a powerful tool in calculus and can help us understand and manipulate complex functions. If you have any further questions, please don't hesitate to ask.