91Ó°ÊÓ

When we think about a moving particle, its velocity tells us how fast it’s moving and in what direction. In terms of parametric equations, the velocity is represented as a vector. This vector is the derivative of the position vector with respect to time. For a given position vector, \( \mathbf{r}(t) = 3 \cos t \mathbf{i} + 2 \sin t \mathbf{j} \), the velocity vector is calculated by differentiating each component with respect to time \( t \).

• Differentiate \( 3 \cos t \) to get \( -3 \sin t \).
• Differentiate \( 2 \sin t \) to get \( 2 \cos t \).

This results in the velocity vector \( \mathbf{v}(t) = -3 \sin t \mathbf{i} + 2 \cos t \mathbf{j} \). Evaluating this at \( t = \frac{\pi}{3} \), we find \( \mathbf{v}\left(\frac{\pi}{3}\right) = -\frac{3\sqrt{3}}{2} \mathbf{i} + 1 \mathbf{j} \). In essence, the velocity vector tells us the speed and direction of the particle's motion at a specific instant in time.

Acceleration indicates how the velocity of a particle is changing over time. In the context of parametric equations, it is the derivative of the velocity vector. For the velocity vector \( \mathbf{v}(t) = -3 \sin t \mathbf{i} + 2 \cos t \mathbf{j} \), we find the acceleration vector by differentiating each component again.

• Differentiate \( -3 \sin t \) to get \( -3 \cos t \).
• Differentiate \( 2 \cos t \) to get \( -2 \sin t \).

The resulting acceleration vector is \( \mathbf{a}(t) = -3 \cos t \mathbf{i} - 2 \sin t \mathbf{j} \). At \( t = \frac{\pi}{3} \), the evaluated vector is \( \mathbf{a}\left(\frac{\pi}{3}\right) = -\frac{3}{2} \mathbf{i} - \sqrt{3} \mathbf{j} \). This provides insight into how the speed and direction of the particle are changing at a given moment. While velocity measures movement, acceleration measures changes in that movement.

The path of a particle offers a visual representation of its trajectory over time. In the example given by the position function \( \mathbf{r}(t) = 3 \cos t \mathbf{i} + 2 \sin t \mathbf{j} \), the path is an ellipse. This means the particle moves in a loop-like motion, traced out by the parametric equations.

• The term \( 3 \cos t \) suggests an x-component of the path with a maximum stretch of 3 units, which is the semi-major axis.
• The term \( 2 \sin t \) indicates a y-component stretch of 2 units, establishing the semi-minor axis.

At \( t = \frac{\pi}{3} \), the particle's position is \( \mathbf{r}\left(\frac{\pi}{3}\right) = \frac{3}{2} \mathbf{i} + \sqrt{3} \mathbf{j} \), marking its location on the ellipse. Drawing the velocity and acceleration vectors from this point helps interpret not just how the particle moves, but clear visual guidance on its changing motion. Through vectors, we grasp both speed direction and how they are modifying over time.