# Velocity in plane polar coordinates

• marcom
In summary, velocity in plane polar coordinates is the rate of change of position described in terms of distance and angle. It can be calculated using the formula v = r * dθ/dt and can be converted to Cartesian coordinates using vx = v * cos(θ) and vy = v * sin(θ). Velocity and speed have different meanings in polar coordinates, with velocity taking into account direction and speed only referring to magnitude. Velocity can also be negative, indicating movement in the opposite direction of the positive direction in polar coordinates.
marcom
Hi,
I have a problem with the following explanation of velocity in plane polar coordinates.

I don't understand why the magnitude of Δer is approximately equal to Δθ.

Thanks

You know from basic geometry that the arc of circumference spanned by ##\Delta \theta## is going to be ##\delta=|\hat{e}_r|\Delta\theta##. However, even if ##\Delta e_r## is actually straight, it is very very small and so it will be close in magnitude to the arc of length. Also since ##\hat{e}_r## has unit magnitude, you find ##\Delta e_r\simeq\delta=\Delta\theta##.

Thanks!

## 1. What is velocity in plane polar coordinates?

Velocity in plane polar coordinates refers to the rate of change of position of an object in a coordinate system that uses polar coordinates. This means that velocity is described in terms of the distance an object moves and the angle at which it moves, rather than simply its x and y coordinates.

## 2. How do you calculate velocity in plane polar coordinates?

To calculate velocity in plane polar coordinates, you can use the formula v = r * dθ/dt, where v is velocity, r is the distance from the origin, and dθ/dt is the angular velocity. You can also use the Pythagorean theorem to find the magnitude of the velocity vector.

## 3. What is the difference between velocity and speed in plane polar coordinates?

Velocity and speed are often confused, but they have different meanings in plane polar coordinates. Velocity refers to the rate of change of position, including the direction an object is moving in. Speed, on the other hand, only refers to the magnitude of the velocity and does not take into account direction.

## 4. How do you convert velocity in plane polar coordinates to Cartesian coordinates?

To convert velocity in plane polar coordinates to Cartesian coordinates, you can use the equations vx = v * cos(θ) and vy = v * sin(θ), where vx and vy are the x and y components of the velocity vector, v is the magnitude of the velocity, and θ is the angle at which the object is moving.

## 5. Can velocity in plane polar coordinates be negative?

Yes, velocity in plane polar coordinates can be negative. Negative velocity means that the object is moving in the opposite direction of the positive direction, which is typically taken to be counterclockwise in polar coordinates. This can also be seen in the negative values for the x and y components of the velocity vector in Cartesian coordinates.

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