Relationship between velocity, acceleration, and a circle?

[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 = v_it + at²/2 (a=acceleration, v_i=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 traveling 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

 

[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

 

[pbuk]

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

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

 

[stvrbbns]

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

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).

 

[stvrbbns]

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

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

 

[mfig]

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

 

[stvrbbns]

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

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).