How Do You Model Motion With Variable Velocity in Two Dimensions?

You model variable-velocity motion in two dimensions by expressing an object's position vector as a function of time, then differentiating its components to obtain the velocity vector and acceleration vector. This is HL-only content from AHL 3.12, Vector applications to kinematics.

The Vector Model

For motion in the Cartesian plane, write

$$\mathbf{r}(t)=\begin{pmatrix}x(t)\\y(t)\end{pmatrix}.$$

The coordinates at time $t$ are ((x(t),y(t))). Motion has variable velocity when at least one velocity component changes with time, causing the object's speed, direction, or both to change.

If acceleration is given, integrate each component to find velocity, then integrate again to find position. Use the initial velocity and initial position to determine every constant of integration.

A useful modelling sequence is: define the time domain, write $\mathbf{r}(t)$, differentiate to obtain $\mathbf{v}(t)$ and $\mathbf{a}(t)$, then evaluate the requested time. If the question asks when the object changes direction or reaches a specified speed, solve the relevant component equation or magnitude equation using the GDC. Keep full precision until the final answer and interpret the result in context.

Worked Example

Suppose

$$\mathbf{r}(t)=\begin{pmatrix}t^2\\t^3-3t\end{pmatrix}\text{ m}.$$

Differentiate component by component:

$$\mathbf{v}(t)=\begin{pmatrix}2t\\3t^2-3\end{pmatrix},\qquad \mathbf{a}(t)=\begin{pmatrix}2\\6t\end{pmatrix}.$$

At $t=2$,

$$\mathbf{v}(2)=\begin{pmatrix}4\\9\end{pmatrix}\text{ m s}^{-1}.$$

Therefore, the speed is

$$\lVert\mathbf{v}(2)\rVert=\sqrt{4^2+9^2}=\sqrt{97}\approx9.85\text{ m s}^{-1}.$$

The velocity is variable because its second component depends on $t$, while the acceleration confirms that the velocity changes over time. Notice that the final speed is a scalar and must not be written as a vector.

Exam Technique

For an HL Paper 1 or Paper 2 response, show the vector function and differentiation before using your GDC, and include units. A common misconception is treating velocity as speed: velocity has magnitude and direction, whereas speed is only its magnitude. Also, intersecting paths do not establish a collision; two objects must occupy the same position at the same time.