Confused by velocity in spherical coordinates

In summary, velocity in spherical coordinates is a measure of an object's speed and direction of motion in three-dimensional space. It is described using three components: radial velocity, tangential velocity, and normal velocity. This representation is more comprehensive compared to other coordinate systems, such as Cartesian coordinates. The formula for converting velocity in spherical coordinates to Cartesian coordinates can be derived using trigonometric identities and basic geometry concepts. Velocity in spherical coordinates is useful in scientific research, particularly in fields like astronomy. It is also related to other kinematic quantities, such as acceleration and displacement, through mathematical relationships.
  • #1
The thinker
56
0
Hello,

I am trying to work out how you derive velocity in terms of spherical coordinates, could anyone point me in the direction of a simple and quite explicit derivation. I keep getting confused!

Thanks.
 
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  • #2
Try http://mathworld.wolfram.com/SphericalCoordinates.html" eq 71-73
 
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FAQ: Confused by velocity in spherical coordinates

1. What is velocity in spherical coordinates?

Velocity in spherical coordinates is a measure of an object's speed and direction of motion at a specific point in space. It is described using three components: radial velocity (Vr), tangential velocity (Vθ), and normal velocity (Vφ). These components represent the velocity in the radial, azimuthal, and polar directions, respectively.

2. How is velocity in spherical coordinates different from other coordinate systems?

In spherical coordinates, velocity is measured using angular components (θ and φ) in addition to the radial component. This makes it a more comprehensive representation of an object's motion compared to other coordinate systems, such as Cartesian coordinates, which only use linear components (x, y, z).

3. What is the formula for converting velocity in spherical coordinates to Cartesian coordinates?

The formula for converting velocity in spherical coordinates (Vr, Vθ, Vφ) to Cartesian coordinates (Vx, Vy, Vz) is as follows: Vx = Vrsinθcosφ, Vy = Vrsinθsinφ, Vz = Vrcosθ. This formula can be derived using trigonometric identities and basic geometry concepts.

4. Why is velocity in spherical coordinates useful in scientific research?

Velocity in spherical coordinates is useful in scientific research because it allows for a more intuitive understanding of an object's motion in three-dimensional space. It is particularly useful in fields such as astronomy, where objects often have complex and non-linear trajectories. Additionally, it can simplify calculations and provide a more accurate representation of an object's velocity compared to other coordinate systems.

5. How is velocity in spherical coordinates related to other kinematic quantities?

Velocity in spherical coordinates is related to other kinematic quantities, such as acceleration and displacement, through basic mathematical relationships. For example, the tangential component of velocity (Vθ) is related to the tangential component of acceleration (aθ) through the equation aθ = Vθ / r, where r is the distance from the object to the origin. Similarly, the displacement in the radial direction (Δr) is related to the radial component of velocity (Vr) through the equation Δr = VrΔt, where Δt is the change in time.

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