# Relationship between velocity, acceleration, and a circle?

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1. Jan 18, 2016

### stvrbbns

The perimeter of a circle is 2πR (R=radius). [ref]
Acceleration = Δv/Δt (v=velocity, t=time). [ref]
Motion mathematics can always be reduced to multiple independent one-dimensional motions. [ref]
The distance an object travels while accelerating = vit + at2/2 (a=acceleration, vi=initial velocity). [ref]
1. If a circle centered at (0,0) has a radius of 2m, then it has a diameter of 4m and a perimeter of 4π m.
2. If an object is moving clockwise around this circle once every 4 seconds, then that object has a speed of 1π m/s.
3. At the top of the circle, the object has an (x,y) velocity of (π,0); let this be t0.
4. At the right of the circle, the object has an (x,y) velocity of (0,-π); let this be t1.
5. Δt = t1-t0 = 1 second.
6. Δv = v1-v0 = (π,0) - (0,-π) = (-π,-π).
7. a = (-π,-π) / second (for that particular time interval).
8. So displacement on the X-axis = π m/s * 1 s + ( (-π m/s2) * (1s)2 / 2 ) = π m + -π/2 m = π/2 ...
But the object travelling the perimeter of the 2m-radius circle every 4 seconds should be at (2,0) 1 second after (0,2). What am I doing wrong - displacement, acceleration, or something else/more?

Thanks.

(other references)
- velocity calculator
- kinematic equations

2. Jan 18, 2016

### Karmaslap

Δv = v1-v0 = (π,0) - (0,-π) = (-π,-π).
This is wrong.
Δv= (π,π). By basic arithmetic, your pies are positive. 0 - - π = π, and π - 0 = π

3. Jan 18, 2016

### MrAnchovy

You can only use the equation $s = ut + \frac 12 a t^2$ when acceleration is constant.

4. Jan 18, 2016

### stvrbbns

Thanks for catching that. I think that it should then be:
Δv = v1-v0 = (0,-π) - (π,0) = (-π,-π)
and that I had the v0 and v1 incorrectly switched; the outcome is correctly still (-π,-π).

5. Jan 18, 2016

### stvrbbns

Ah, I was trying to take the average acceleration and treat it as constant.

6. Jan 19, 2016

### mfig

Plots of the actual component numbers show just how off an assumption of constant acceleration for each component is for this problem.

Last edited: Jan 19, 2016
7. Jan 19, 2016

### stvrbbns

Thank you very much! That relationship between position, velocity, and acceleration exposes quite a bit of my problem.

[ref] For anyone reading this post later, acceleration is the derivative of velocity and velocity is the derivative of position. The graphs posted by @spamanon show all 3 of those for my circle (red position, blue velocity, black acceleration).