Unit Vectors in Motion : Analyzing Their Time Derivatives 😮| Velocity in Polar Coordinates
Channel: SIGMA PHYSICS ⚛️ 
Date: 28/05/2023

PreRequisetes:
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https://www.youtube.com/playlist?list=PLQ3CKuXeJ7_O5VP3IDZxV8f3eEjVjBCTE
and
https://www.youtube.com/playlist?list=PLQ3CKuXeJ7_OES9pM2O_ra5No4LXe3fU7

In polar coordinates, the position of a point is described by the radial distance (r) from the origin and the angle (θ) it makes with a reference direction (usually the positive x-axis in Cartesian coordinates). To describe the velocity and acceleration of a point in polar coordinates, we need to consider the time derivatives of the radial and angular quantities.

1. Velocity in Polar Coordinates:
The velocity of a point in polar coordinates is a vector that describes the rate of change of position. It consists of two components: the radial velocity (vᵣ) and the tangential velocity (vₜ).

The radial velocity (vᵣ) represents the rate of change of the radial distance (r) with respect to time (t). It can be calculated using the chain rule:

vᵣ = dr/dt

The tangential velocity (vₜ) represents the rate of change of the angular position (θ) with respect to time (t) multiplied by the radial distance (r). It is given by:

vₜ = r dθ/dt

Therefore, the velocity vector in polar coordinates can be expressed as:

v = vᵣ * ẑᵣ + vₜ * ẑₜ

Here, ẑᵣ is the unit vector in the radial direction, and ẑₜ is the unit vector in the tangential direction.

2. Acceleration in Polar Coordinates:
The acceleration of a point in polar coordinates describes the rate of change of velocity. Similar to velocity, it consists of two components: the radial acceleration (aᵣ) and the tangential acceleration (aₜ).

The radial acceleration (aᵣ) represents the rate of change of the radial velocity (vᵣ) with respect to time (t). It can be calculated using the chain rule:

aᵣ = dvᵣ/dt = d²r/dt²

The tangential acceleration (aₜ) represents the rate of change of the tangential velocity (vₜ) with respect to time (t) multiplied by the radial distance (r), plus the square of the tangential velocity divided by the radial distance. It is given by:

aₜ = (r d²θ/dt²) - (vₜ² / r)

Therefore, the acceleration vector in polar coordinates can be expressed as:

a = aᵣ * ẑᵣ + aₜ * ẑₜ

3. Time Derivative of Unit Vectors:
The unit vectors in polar coordinates, ẑᵣ and ẑₜ, are direction vectors that change with time due to the changing angle (θ). Their time derivatives can be calculated as follows:

dẑᵣ/dt = d/dt(cosθ ẑₓ + sinθ ẑᵧ)
        = -sinθ(dθ/dt)ẑₓ + cosθ(dθ/dt)ẑᵧ
        = -sinθ(dθ/dt)ẑᵣ + cosθ(dθ/dt)ẑₜ

dẑₜ/dt = d/dt(-sinθ ẑₓ + cosθ ẑᵧ)
        = -cosθ(dθ/dt)ẑₓ - sinθ(dθ/dt)ẑᵧ
        = -cosθ(dθ/dt)ẑᵣ - sinθ(dθ/dt)ẑₜ

The time derivatives of the unit vectors account for the change in direction with respect to time as the angle (θ) changes.



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