Module 5.5 · Lesson 5.5.2

Vector-Valued Functions — Position, Velocity & Acceleration

Watch position, velocity, and acceleration vectors update frame-by-frame along a parametric path, exactly as an animation engine advances a moving object.

Intuition

A vector-valued function $\mathbf{r}(t)=\langle x(t),y(t)\rangle$ hands back a position vector at every instant $t$. Differentiate once and you obtain the velocity vector $\mathbf{v}(t)=\mathbf{r}'(t)$; differentiate again and you obtain the acceleration $\mathbf{a}(t)=\mathbf{r}''(t)$. Each new frame of an animation simply evaluates these three vectors at the next time stamp.

Bézier curves used in motion graphics are built the same way: the control points define a parametric path, and the animation system samples $\mathbf{r}(t)$, $\mathbf{v}(t)$, and $\mathbf{a}(t)$ to place, orient, and ease the object.

Core concept

Position is the vector $\mathbf{r}(t)$. Velocity is its derivative—the tangent that points in the direction of motion and whose length is speed. Acceleration is the derivative of velocity; it tells how the velocity vector itself is changing length or direction.

$\mathbf{r}(t)$$\langle t,\,t^{2}\rangle$
At $t=1$$\mathbf{r}=\langle 1,1\rangle$
$\mathbf{v}=\mathbf{r}'$$\langle 1,2t\rangle=\langle 1,2\rangle$
$\mathbf{a}=\mathbf{r}''$$\langle 0,2\rangle$
Speed$|\mathbf{v}|=\sqrt{5}$

Interactive Lab

Drag the time slider. The amber point traces the path; the green arrow is velocity and the cyan arrow is acceleration, both redrawn every frame.

Speed $|\mathbf{v}|$: 2.24
x y
$\mathbf{r}(t)=\langle t,\,t^{2}\rangle$

Green arrow = velocity (tangent, length = speed). Cyan arrow = acceleration (here constant $\langle 0,2\rangle$). This is the same data an animation system reads each frame to move an object along a Bézier path.

Key Idea

$$ \mathbf{r}(t)=\langle x(t),y(t)\rangle,\quad \mathbf{v}(t)=\mathbf{r}'(t),\quad \mathbf{a}(t)=\mathbf{r}''(t) $$

A vector-valued function packages the coordinates into a single object that can be differentiated component-wise. The first derivative is the velocity vector; the second is the acceleration vector. Speed is the magnitude $|\mathbf{v}(t)|$. In animation and physics engines these three vectors are evaluated at successive time steps to advance the object along its path.

$\mathbf{r}(t)$Position vector at time $t$
$\mathbf{v}(t)$Velocity = first derivative (direction of motion)
$\mathbf{a}(t)$Acceleration = second derivative
$|\mathbf{v}|$Speed (length of the velocity vector)

Think Further

  1. If acceleration is always perpendicular to velocity, what is happening to the speed of the object?
  2. In a cubic Bézier animation curve, why does the velocity vector change length even when the object appears to move at “constant speed” on screen?
  3. How would you write the unit tangent vector in terms of $\mathbf{v}(t)$, and why is it useful for orienting a sprite along its path?