91Ó°ÊÓ

In polar coordinates, the velocity vector describes how quickly an object moves and the direction it travels along the radial and angular axes. The velocity vector consists of two components: the radial velocity, denoted as \( v_r \), and the angular velocity, denoted as \( v_\theta \). The radial velocity \( v_r \) reflects how fast the object is moving towards or away from the origin, while the angular velocity \( v_\theta \) indicates how fast the object is going around the origin.

To find these components, we differentiate the radial position function \( r = e^{a\theta} \) with respect to time \( t \).

• Radial Velocity: We use the chain rule, obtaining \( \frac{dr}{dt} = \frac{dr}{d\theta} \cdot \frac{d\theta}{dt} \), which simplifies to \( v_r = 2ae^{a\theta} \).
• Angular Velocity: It is found by multiplying the radial position by the rate of change of \( \theta \), so \( v_\theta = r \cdot \frac{d\theta}{dt} = 2e^{a\theta} \).

Finally, the velocity vector \( \mathbf{v} \) is expressed in terms of the basis vectors \( \mathbf{u}_r \) and \( \mathbf{u}_\theta \) as \( \mathbf{v} = 2ae^{a\theta} \mathbf{u}_r + 2e^{a\theta} \mathbf{u}_\theta \). This defines the overall speed and movement direction in the polar coordinate system, combining how fast an object moves radially and angularly.

The acceleration vector in polar coordinates reveals how an object's velocity changes over time. Similar to velocity, acceleration comprises radial and angular components. The radial acceleration \( a_r \) shows the rate of change of the radial speed, while the angular acceleration \( a_\theta \) indicates the rate at which the angular speed changes.

To compute these components, we differentiate the velocity expressions obtained earlier.

• Radial Acceleration: By differentiating the radial velocity \( v_r = 2ae^{a\theta} \), we find \( a_r = 4a^2e^{a\theta} \).'
• Angular Acceleration: It is given by \( a_\theta = r \cdot \frac{d^2\theta}{dt^2} + 2 \cdot \frac{dr}{dt} \cdot \frac{d\theta}{dt} \). Since \( \frac{d^2\theta}{dt^2} = 0 \), the formula simplifies to \( a_\theta = 8ae^{a\theta} \).

The complete acceleration vector \( \mathbf{a} \) in polar coordinates is written as \( \mathbf{a} = 4a^2e^{a\theta} \mathbf{u}_r + 8ae^{a\theta} \mathbf{u}_\theta \). Understanding these components helps determine how an object's speed and direction evolve in time in the context of a polar reference frame.

Differentiation plays a crucial role in calculating velocity and acceleration in polar coordinates. It is the mathematical process used to find the rate at which one quantity changes with respect to another, which is essential in dynamics to understand motion.

• By differentiating the radial position \( r = e^{a\theta} \) with respect to \( t \), we determine how quickly the distance from the origin changes over time, giving us the radial velocity \( v_r \). This applies the chain rule: first differentiating \( r \) with respect to \( \theta \) and then multiplying by \( \frac{d\theta}{dt} \).
• Similarly, differentiating the velocity components leads us to the acceleration components. This continues to use the chain rule, as seen in finding \( a_r \), and ensures all temporal dynamics are accurately captured.

This process of differentiation, particularly using the chain rule, demonstrates how calculus facilitates the transition from static functions to dynamic motions, making it foundational in the context of understanding vector calculus in polar coordinates.