Velocity and all of the good stuff that goes along with it.

[jasonlr82794]

Hey, I was wondering if you guys could give me some input in how you view velocity and what it means to you guys. I already have a rough definition in my head but I also think it would be quite helpful if I had some different input on the subject. Also, applications in real life would help as well. Thank you.

 

[Vorde]

Velocity is defined as $\frac{\Delta Displacement}{\Delta Time}$ or $\frac{dx}{dt}$ or anyone of probably dozens of equivalent definitions. What are you looking for?

 

[jasonlr82794]

Im looking for instantaneous velocity which Is what you put on the post. I understand the math behind finding instantaneous velocities but what doesn't make sense is with an instantaneous velocity it is the velocity at a given instant. with average velocity to find the velocity you take the average of two points and that will always be your velocity on a linear graph. On a every changing velocity graph such as x squared you have your positions but when finding this instant velocity you don't use two points but a limit. It just doesn't make sense. How was the idea of an instant. velocity conceived? I know all of the math behind it but the conceptual part is what's getting me. How did you learn this subject?

 

[Mandelbroth]

Velocity is defined as $\frac{\Delta Displacement}{\Delta Time}$ or $\frac{dx}{dt}$ or anyone of probably dozens of equivalent definitions. What are you looking for?

If I may be pedantic, I believe you mean $\frac{d\vec{x}}{dt}$.

You seem to be looking at it in a Zeno's Paradox-like way. Instantaneous velocity is a lot like the average velocity between two infinitesimally distant points in time.

 

[Vorde]

If I may be pedantic, I believe you mean $\frac{d\vec{x}}{dt}$.

You seem to be looking at it in a Zeno's Paradox-like way. Instantaneous velocity is a lot like the average velocity between two infinitesimally distant points in time.

You may be :P

Totally agree with this, it turns out that if you look at the average velocity over two points and you keep making the points closer to each other, the average velocity keeps getting closer and closer to a specific number. If you make the points infinitely close, the average velocity gets infinitely close to this number. This number is the instantaneous velocity.

 

[jasonlr82794]

What is zenos paradox? and you know how for the average velocity you have y/x? Well the derivative for x squared is 2x so how would I put that in the y/x form? where does the y come into the 2x. They both change with each other so how would you represent that with 2x but more in depth?

 

[Mandelbroth]

What is zenos paradox? and you know how for the average velocity you have y/x? Well the derivative for x squared is 2x so how would I put that in the y/x form? where does the y come into the 2x. They both change with each other so how would you represent that with 2x but more in depth?

I like to consider myself a pure mathematician for the most part, so I prefer using covectors to define derivatives so that no one knows what I'm talking about except people who know stuff. But that's messy.

For the more common usage, we consider the derivative of a scalar function of one variable by finding a limit of the difference quotient, given by
$$\lim_{\Delta x\rightarrow 0}\frac{y(x+\Delta x)-y(x)}{\Delta x} = \frac{dy}{dx}$$

To get 2x using the limit method,

$$\frac{d}{dx}[x^2]=\lim_{\Delta x\rightarrow 0}\frac{(x+\Delta x)^2-x^2}{\Delta x} = \lim_{\Delta x\rightarrow 0}\frac{x^2+2x\Delta x +(\Delta x)^2-x^2}{\Delta x} = \lim_{\Delta x\rightarrow 0}(2x+\Delta x) = 2x$$

Velocity is different, because it's a vector, but in one dimension (forward and backward movement only) we can treat it as a scalar.

Zeno's paradoxes are a bunch of famous paradoxes that were conceived by a Greek philosopher.