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Say the position of a particle is expressed in polar coordinates as

([tex]\phi[/tex],r)

If we want to describe it's velocity

**v**we need to differentiate both components(angular and radial) with respect to time, as well as adding a directional component and giving each magnitude(speed(?)).

So, differentiating individual components we say rate of change of the radial component is

[tex]\frac{dr}{dt}[/tex]

and hence velocity of radial component = [tex]\dot{r}[/tex][tex]\hat{r}[/tex]

where we define [tex]\hat{r}[/tex] to be the unit vector pointing outwards in the positive direction along the radial component.

secondly we need to define the rate of change of the angular component

we can say this is [tex]\frac{d\theta}{dt}[/tex]

and to give it direction we define [tex]\hat{\theta}[/tex] to be the unit vector pointing perpendicular to the radial line in the counterclockwise direction.

now here is the more delicate bit: we need to add a factor of r to this in order to give the angular motion a magnitude, otherwise particles close to the origin would be moving at the same velocity as those far from it. This naturally only works if we are using radians.

hence the final velocity is described as

[tex]v=r\hat{r}+r\dot{\theta}\hat{\theta}[/tex]

So what we've done is shifted from polar to vectorial system with the vector components of the velocity at the position of the particle at any time, adding to give the speed and direction.

I may post this in other forums since it falls under more than one category, thanks in advance.