Can you explain the basics of Newton's law?

[Herbststurm]

Hello

Maybe I am standing on a garden hose (great German adage, does it exist in english or are you confused what I am talking about?)

Okay, to the topic:
Just for one dimension. It is easier:

We have m\ddot{x} = F(x(t)) Now my book tells me that if I expansion with the velocity I should get:

\frac{m}{2} \frac{d}{dt} \dot{x}^{2} = - \frac{d}{dt} U(x(t))

I don't understand. Sadly I learnet it a year ago and I forgot everything. How sad a fate! :-(

I understand the right-hand side. Force is the derivative of the potential, the field must be conservative. But I don't understand the left-hand side.

1.) why is there a two under the mass?

2.) I expansion with the velocity? the differentialoperator on the velocity is my acceleration and the second velocity is one from the expansion. Where is the second? I could only identify one velocity. What happened?

Thanks
greetings

 

[atyy]

v=dx/dt; dv/dt=d²x/dt²

d(v²)/dt
=[d(v²)/dv][dv/dt]
=[2v][dv/dt]
=[2dx/dt][dv/dt]

Rearrange using first and last lines of the above:
dv/dt=[d(v²)/dt][dt/dx]/2

LHS: m[dv/dt]=m[d(v²)/dt][dt/dx]/2

RHS: F=-dU/dx=-[dU/dt][dt/dx]

LHS=RHS
m[d(v²)/dt][dt/dx]/2=-[dU/dt][dt/dx]
m[d(v²)/dt]/2=-[dU/dt]

 

[Dale]

Maybe I am standing on a garden hose (great German adage, does it exist in english or are you confused what I am talking about?)

I work with a lot of Germans, so I would really like to know what "standing on a garden hose" means.

 

[Herbststurm]

w = \int \vec{F} ~ d\vec{r} = m ~ \int \vec{a} ~ d\vec{r}

\vec{a} = \frac{d \vec{v}}{dt} = \frac{d \vec{v}}{dt} \frac{d \vec{r}}{d\vec{r}} = \frac{d \vec{v}}{dr} \frac{d \vec{r}}{dt} = \frac{d \vec{v}}{dr} \vec{v}

\Rightarrow w = m ~ \int \frac{d \vec{v}}{dr} \vec{v} ~ d\vec{r} = m ~ \int d \vec{v} ~ \vec{v} = m \frac{v^{2}}{2} = \frac{1}{2} m v^{2}

Okay, I got it again :)

Damn, it is bad how fast on is able to forgot things

I work with a lot of Germans, so I would really like to know what "standing on a garden hose" means.

It means that on has a huge black out. If you know something in general but you forgot it in this moment. Than you are standing on a garden hose.

greetings

 

[redargon]

Hello

Maybe I am standing on a garden hose (great German adage, does it exist in english or are you confused what I am talking about?)

Is that similar to going down the wooden track (auf dem holzen Bahn oder etwas? Es is eine lange Zeit sinds ich deutsch geschrieben habe)

 

[Herbststurm]

Is that similar to going down the wooden track (auf dem holzen Bahn oder etwas? Es is eine lange Zeit sinds ich deutsch geschrieben habe)

Hi,

no it is not similar. What you mean is "Sich auf dem Holzweg befinden". This means that somebody has a idea but the idea is wrong. If you do wrong calculations or assumptions than you are going down the wooden track.
The garden hose means black out, not wrong ideas.

greetings

 

[Dale]

It means that on has a huge black out. If you know something in general but you forgot it in this moment. Than you are standing on a garden hose.

That makes sense, so the idea is somewhere backed up in the garden hose, but since you are standing on it the idea won't come out.

I think we would say "I lost my train of thought" meaning the thought is there on the tracks somewhere, and you just cannot find it right now.

 

[disregardthat]

In Norway, that saying is equivalent to saying that you're having an "iron curtain". Dunno if this is used in other countries as well, but it refers to the information block in Europe during the cold war.