Speed versus velocity?

[The Badger]

Velocity is how fast something travels in a particular direction.

Does that direction have to be in specific terms? Or can you just say, "I'm going to drive over there at 50 mph."? Would that count as velocity?

It depends on how you define 'direction'.

Example one: You are in a car, and define 'straight ahead' as your positive direction. You drive forwards at 50mph. Your speed is 50mph, your velocity is +50mph. You stop, and reverse at 10mph. Your speed is 10mph, your velocity is -10mph.

Example two. A spot has been painted on the surface of a straight road. Traveling towards that spot is considered positive, traveling away is negative. You set off, straight towards it, at 50mph. Your speed is 50mph, your velocity is +50mph. You do not stop or alter your speed in anyway. After passing over the spot you are now traveling away from it. Your speed is unchanged at 50mph, but your velocity is now -50mph.

Example three. You are now traveling in a perfectly circular course around the spot from the previous example. Again, approaching it is considered positive, traveling away is negative. You maintain a constant speed of 50mph, but because of your course you get neither nearer or farther from the spot. Hence, your speed is 50mph, and your velocity is 0mph.


So, yes, RoJoHen, 'over there' could be considered a direction.

 

[sojourner]

^Good examples!

 

[Jadzia]

What happens in polar coordinates, if r'=0 and θ'>0 and both constant.

I honestly don't know if that's considered to be accelerating or not.

Because on one hand, we can take a purely mathematical perspective of acceleration as being when double derivatives of positional coordinates are non zero, which is obviously coordinate dependent. I like this definition.

On the other hand, you can take the physics route and think of acceleration in terms of Newtonian mechanics which formulates around cartesian coordinates. This brings something new to the equation; the fake forces called centrifugal and coriolis, that appear in the transformations to curvilinear coordinate systems.

So on one hand we could say acceleration in the radial direction is r'', and on the other hand say it is r'' - rθ'^2.