How Do You Model Linear Motion With Constant Velocity Using Vectors?

An object moving with constant velocity is modelled by $\mathbf{r}(t)=\mathbf{r}_0+t\mathbf{v}$, where $\mathbf{r}_0$ is its initial position and $\mathbf{v}$ is its constant velocity. The parameter $t$ represents elapsed time.

This is linear motion because neither the direction nor magnitude of the velocity changes. It belongs to AHL 3.12: Vector applications to kinematics, so it is IB Mathematics: Applications and Interpretation HL-only content.

The Mechanism

The model adds the displacement $t\mathbf{v}$ to the initial position vector. Since multiplying $\mathbf{v}$ by $t$ changes its magnitude but not its direction, the object follows a straight path. Units must be consistent: if position is measured in kilometres and velocity in kilometres per hour, then $t$ must be measured in hours.

Suppose a drone begins at $(2,5,1)$ km and travels with velocity $(3,-1,2)$ km h$^{-1}$. Its position vector is

$$\mathbf{r}(t)=\begin{pmatrix}2\\5\\1\end{pmatrix}+t\begin{pmatrix}3\\-1\\2\end{pmatrix}.$$

After $4$ hours,

$$\mathbf{r}(4)=\begin{pmatrix}2\\5\\1\end{pmatrix}+4\begin{pmatrix}3\\-1\\2\end{pmatrix}=\begin{pmatrix}14\\1\\9\end{pmatrix}\text{ km}.$$

In component form, $x=2+3t$, $y=5-t$, and $z=1+2t$. These parametric equations show how each coordinate changes with time. The is the magnitude of the velocity:

$$\lvert\mathbf{v}\rvert=\sqrt{3^2+(-1)^2+2^2}=\sqrt{14}\approx3.74\text{ km h}^{-1}.$$

Over the four-hour interval, the displacement is $4\mathbf{v}=(12,-4,8)$. Notice that displacement is a vector, whereas distance is a scalar. For this constant straight-line motion with no reversal, the distance travelled is $4\sqrt{14}$ km. This provides a useful check against the change from initial to final position and confirms that the velocity magnitude has been applied correctly.

A common misconception is that intersecting paths guarantee a collision. They do not: two objects collide only when their position vectors are equal at the same value of $t$. Equate corresponding components and check that one common, physically valid time satisfies every equation.

Exam Technique

For HL Paper 1 or Paper 2, state the model and units before substituting. Show the component equations when solving for position or collision time, use the GDC where appropriate, and interpret the result in context.