Velocity & Relative Velocity

The most-feared corner of the vectors topic, and the most routine once one modelling line becomes habit:

$\mathbf{r}(t) = \mathbf{r}_0 + t\mathbf{v}$, position at time $t$ = initial position + $t \times$ velocity

Write this for each moving object before doing anything else. It converts the word problem into component algebra, and the written models are the first method marks.

Interception and collision

At 12:00, ship $A$ is at $(2\mathbf{i} + 3\mathbf{j})$ km with velocity $(4\mathbf{i} + \mathbf{j})$ km/h; ship $B$ at $(14\mathbf{i} - 3\mathbf{j})$ km with velocity $(\mathbf{i} + 2.5\mathbf{j})$ km/h. Show they meet. $\mathbf{r}_A(t) = (2 + 4t)\mathbf{i} + (3 + t)\mathbf{j}$; $\mathbf{r}_B(t) = (14 + t)\mathbf{i} + (-3 + 2.5t)\mathbf{j}$ Equate $\mathbf{i}$: $2 + 4t = 14 + t$ $\to$ $t = 4$ Check $\mathbf{j}$ at $t = 4$: $3 + 4 = 7$ and $-3 + 10 = 7$ ✓ $\to$ they meet at 16:00

The $\mathbf{j}$-check is not optional: one component agreeing proves nothing, the same $t$ must satisfy both, and the verification line carries the “show” mark. If the $\mathbf{j}$-components disagree, the correct answer is “they do not collide”, with the numbers shown.

Relative velocity and relative position

Velocity of $A$ relative to $B$: $\mathbf{v}_A - \mathbf{v}_B$, how $A$ moves as seen from $B$. Relative position likewise: $\mathbf{r}_A - \mathbf{r}_B$. Two standard uses: “show the ships do not meet” (relative position never zero), and distance questions, the distance apart at time $t$ is $|\mathbf{r}_A(t) - \mathbf{r}_B(t)|$, minimised either by completing the square on the squared distance or noting closest approach geometry. Subtraction order matters: “A relative to B” is $A$ minus $B$; reversing it negates the picture.

Courses, currents and winds

A swimmer aims across a river; the current drags downstream: actual velocity = still-water velocity + current vector. Draw the triangle, label all three vectors, and the question reduces to trigonometry or Pythagoras, heading questions (“what course must the pilot set…”) solve the triangle for the aimed vector instead. The unlabelled-diagram attempt is the main cause of sign chaos here; the labelled triangle is itself creditable working.

Common mistakes

• No position models written, algebra attempted from prose
• Interception “shown” on one component
• $\mathbf{v}_A - \mathbf{v}_B$ reversed
• Speed used where the velocity vector is needed (the unit-vector construction)
• Course triangles drawn without labels, then angles guessed

Full topic context: Vectors notes, note the contrast with calculus kinematics: here velocity is constant; there it varies.