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!)