# Relationship between velocity, acceleration, and a circle?

• stvrbbns
In summary, the perimeter of a circle is 2πR and the distance an object travels while accelerating is given by the equation vit + at^2/2. These equations can be used to calculate the displacement, velocity, and acceleration of an object moving in a circular motion. However, it is important to note that these equations can only be used when acceleration is constant, which may not always be the case.
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 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

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

stvrbbns
stvrbbns said:
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
Karmaslap said:
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).

MrAnchovy said:
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.

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

Last edited:
stvrbbns
spamanon said:
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).

## 1. How does velocity affect the motion of an object in a circular path?

Velocity is a vector quantity that describes the rate of change of an object's position in a specific direction. In a circular motion, velocity determines the speed at which an object moves around the circle, as well as the direction it is moving in at any given point. As the velocity changes, the object's position and direction of motion also change, resulting in a circular path.

## 2. What is the relationship between acceleration and velocity in a circular motion?

Acceleration is defined as the rate of change of an object's velocity. In a circular motion, acceleration is constantly acting towards the center of the circle, which is known as centripetal acceleration. The magnitude of the centripetal acceleration is directly proportional to the square of the object's velocity and inversely proportional to the radius of the circle.

## 3. Can an object have a constant velocity and still be accelerating in a circular motion?

Yes, an object can have a constant velocity and still be accelerating in a circular motion. This is because acceleration is a vector quantity that takes into account the direction of motion. In a circular motion, even if the object is moving at a constant speed, its direction of motion is constantly changing, resulting in a non-zero acceleration towards the center of the circle.

## 4. How does the radius of a circle affect the velocity and acceleration of an object in circular motion?

The radius of a circle is directly proportional to the velocity of an object in circular motion. This means that as the radius of the circle increases, the velocity of the object also increases. However, the radius is inversely proportional to the acceleration, meaning that as the radius increases, the acceleration decreases. This is because a larger circle requires less acceleration to maintain the same velocity as a smaller circle.

## 5. Is there a maximum velocity an object can have in circular motion?

No, there is no maximum velocity an object can have in circular motion. The velocity of an object in circular motion depends on the radius of the circle and the magnitude of the centripetal acceleration. As long as the acceleration is able to keep the object moving in a circular path, the velocity can continue to increase. However, there may be practical limitations such as the strength of the object or the force providing the acceleration that can ultimately limit the maximum velocity.

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