Velocity in spherical coordinates

In summary, velocity in spherical coordinates is a method of describing the movement of an object in three-dimensional space using a radial distance, an angle from the z-axis, and an angle from the x-axis. It is calculated using the conversion equations between spherical and Cartesian coordinates, and the velocity vector is determined by taking the derivative of the position vector with respect to time. Using spherical coordinates simplifies the calculation of velocity for objects moving in circular or spherical paths and allows for a better understanding of direction and magnitude. It can also be converted to other coordinate systems and has various applications in physics, engineering, navigation, robotics, and computer graphics.
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Why the velocity in spherical coordinates equal to ## v^2 = v \dot{} v = \dot{r}^2 + \dot{r}^2\dot{\theta}^2##

maybe

## v^2 = v \dot{} v = (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} + \hat{ \phi } \dot{\phi } r \sin{ \theta}) \dot{} (\hat{ \theta } \dot{ \theta } r +\hat{r} \dot{r} + \hat{ \phi } \dot{\phi } r \sin{ \theta}) = \dot{\theta}^2 r^2 + \dot{r}^2 + r^2 \dot{\phi}^2 \sin^2{\theta} ##

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  • #2
The movement is in a plane, so they've picked coordinates such that the plane is the equatorial plane of the coordinates - hence ##\dot{\phi}=0##.

Incidentally, it should be ##r^2{\dot{\theta}}^2## in your first expression. I presume that's just a typo since it's correct in the text and your next expression.
 
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  • #3
The question about why the velocity in spherical coordinates takes the form it does has been answered, so any further discussion of this problem should be in a new thread in the homework forums.

This thread can be closed.
 

1. What is the formula for calculating velocity in spherical coordinates?

The formula for calculating velocity in spherical coordinates is v = (ṙsinθcosϕ + rθ̇cosθcosϕ - rϕ̇sinθsinϕ)er̂ + (ṙsinθsinϕ + rθ̇cosθsinϕ + rϕ̇sinθcosϕ)eθ̂ + (ṙcosθ - rθ̇sinθ)eϕ̂, where ṙ, θ̇, and ϕ̇ represent the time derivatives of the spherical coordinates and er̂, eθ̂, and eϕ̂ represent the unit vectors in the radial, polar, and azimuthal directions, respectively.

2. How is velocity in spherical coordinates different from velocity in Cartesian coordinates?

Velocity in spherical coordinates is different from velocity in Cartesian coordinates because it takes into account the changing direction of the unit vectors in the radial, polar, and azimuthal directions. In Cartesian coordinates, the unit vectors are constant, whereas in spherical coordinates, they change with respect to the changing angles θ and ϕ.

3. What is the significance of the radial component in the formula for velocity in spherical coordinates?

The radial component in the formula for velocity in spherical coordinates represents the speed of the object in the direction of the radius from the origin. This component is affected by the change in the radial distance ṙ, which can be caused by changes in the magnitude or direction of the velocity.

4. How do you convert velocity in spherical coordinates to velocity in Cartesian coordinates?

To convert velocity in spherical coordinates to velocity in Cartesian coordinates, you can use the following equations: vx = v sinθ cosϕ + vθ cosθ cosϕ - vϕ sinθ sinϕ, vy = v sinθ sinϕ + vθ cosθ sinϕ + vϕ sinθ cosϕ, and vz = v cosθ - vθ sinθ, where v represents the magnitude of the velocity in spherical coordinates.

5. Can velocity in spherical coordinates be negative?

Yes, velocity in spherical coordinates can be negative. This can occur when the object is moving in the opposite direction of the unit vectors in the radial, polar, and azimuthal directions. For example, if the object is moving in the negative r direction, the radial component of the velocity will be negative.

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