In parametric motion, velocity and acceleration are closely linked because of how we define them.

• Velocity is how fast something moves and in what direction. For an object moving along paths described by equations $x(t)$ and $y(t)$, the velocity can be shown with a vector $\mathbf{v}(t)$, which is written like this:

  $$\mathbf{v}(t) = \left( \frac{dx}{dt}, \frac{dy}{dt} \right)$$

• Acceleration is how quickly the velocity itself is changing. We can express acceleration with a vector $\mathbf{a}(t)$ like this:

  $$\mathbf{a}(t) = \left( \frac{d^2x}{dt^2}, \frac{d^2y}{dt^2} \right)$$

From these definitions, we can see that acceleration is really the change in the velocity vector over time.

• The Connection: The acceleration vector can be seen as the change in the velocity vector. If we think of velocity as $\mathbf{v}(t)$, we can write:

  $$\mathbf{a}(t) = \frac{d\mathbf{v}}{dt}$$

This means that any changes in the velocity vector—like moving faster or turning—will directly affect the acceleration of the object that’s moving.

• Why It Matters: So, when we look at motion in the plane using parametric equations, understanding how velocity changes over time helps us learn more about acceleration and how things move overall.