Understanding Velocity in Polar Coordinates

In summary, when deriving the velocity equation of motion in polar coordinates, the position is given by R times R hat instead of theta times theta hat plus R times R hat. This is because theta times theta hat does not have units of length and cannot be added to R times R hat.
  • #1
srmeier
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0

Homework Statement



I don't understand why when we derive the velocity equation of motion in polar coordinates we start with position equal to R times R hat and not (theta times theta hat + R times R hat).

Homework Equations



none really..

The Attempt at a Solution



Is there an assumption I'm missing? or is it simply differentiating linear and angular velocity that is messing me up?
 
Last edited:
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  • #2
It's because your position is given by

[tex]\vec{R} = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} R\cos\theta \\ R\sin\theta \end{pmatrix} = R\begin{pmatrix} \cos\theta \\ \sin\theta \end{pmatrix} = R \hat{R}[/tex]

Also, if you think about it, [tex]\theta \hat{\theta}[/tex] doesn't have units of length, so it's not a displacement and you can't add it to [tex]R\hat{R}[/tex].
 

What is velocity in polar coordinates?

Velocity in polar coordinates refers to the measurement of an object's speed and direction in relation to a fixed origin point. It is typically represented by two components: the radial velocity, which is the speed towards or away from the origin, and the tangential velocity, which is the speed in a circular direction around the origin.

How is velocity in polar coordinates calculated?

Velocity in polar coordinates is calculated using a combination of trigonometry and calculus. The radial velocity is determined by taking the derivative of the radial distance with respect to time, while the tangential velocity is found by multiplying the angular velocity with the radial distance.

What are the advantages of using polar coordinates to measure velocity?

One advantage of using polar coordinates is that it allows for a more intuitive representation of circular motion. Additionally, it can be easier to visualize and analyze complex motion in polar coordinates compared to Cartesian coordinates.

Can velocity in polar coordinates be converted to velocity in Cartesian coordinates?

Yes, velocity in polar coordinates can be converted to velocity in Cartesian coordinates using equations that relate the two coordinate systems. For example, the radial and tangential velocities in polar coordinates can be converted to the x and y components of velocity in Cartesian coordinates.

What real-life applications use velocity in polar coordinates?

Veloctiy in polar coordinates is commonly used in fields such as astronomy, where it is used to track the motion of celestial bodies. It is also used in navigation and robotics to determine the movement of objects in circular paths. Additionally, polar coordinates are used in radar and sonar systems to detect and track the location and velocity of objects.

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