Parametric equations for the position of an object are...

Parametric equations for the position of an object are given. Find the object's velocity and speed at the given times and describe its motion.
$$\left\{\begin{array}{l}x=2 \cos t \\ y=3 \sin t\end{array}\right.$$
(a) $t=0$
(b) $t=\frac{\pi}{2}$

Key Concepts

Speed Calculation

The speed of an object is the magnitude of its velocity vector. By combining the squared components of the velocity and taking the square root, we obtain a scalar quantity that describes how fast the object is moving, regardless of its direction.

Velocity Vector

The velocity vector is obtained by differentiating the position vector with respect to time. It indicates both the speed (magnitude) and the direction of the object's motion. Computing the velocity vector is fundamental in analyzing motion in any direction.

Motion Description

Understanding the velocity and speed helps in describing the overall motion of the object. This involves characterizing the path (such as an ellipse in this case) and determining the nature of the motion at specific times by examining the velocity vector and its magnitude.

Parametric Equations

Parametric equations represent the coordinates of a moving object as functions of a parameter, typically time. This approach lets us describe complex motions, such as curves that are not functions of x or y alone, by linking both coordinates to an independent variable.

Derivatives in Parametric Form

Taking the derivative of each coordinate with respect to time is essential in studying motion. This process uses the chain rule to differentiate the parametric equations, resulting in the components of the velocity vector, which describe how the position changes over time.

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