Foundations — Relative motion — 1D and 2D; river-boat problems

This page assumes you have seen nothing. We build every arrow, letter, and subscript the parent note river-boat topic leans on, in an order where each idea rests on the one before it. By the end of §7 you will have derived the famous subtraction rule yourself; we hold off writing it in symbols until you own every piece.

1. What is a "vector"? (an arrow with a length and a direction)

Before any physics, we need the object that carries velocity: the vector.

We write a vector with a little arrow on top: $\vec{v}$. The plain letter $v$ (no arrow) means only its length, called the magnitude — always a positive number or zero.

Read the figure carefully. The solid burnt-orange arrow near the bottom-left is $\vec{v}$; the teal double-headed label beside it marks its length $=$ magnitude $v$, and the plum note marks its tilt $=$ direction. Now look at the faded orange arrow up and to the right: it was made by sliding the solid one — same length, same tilt, drawn in a new spot. The figure's whole point is that both faded and solid arrows are the same vector $\vec{v}$: position on the page carries no meaning, only length and direction do.

Why the topic needs this: a boat's velocity, the current's velocity, and what the ground observer sees are all "how much + which way" — exactly arrows. See Vectors — addition, components, unit vectors to go deeper.

2. Components — cutting an arrow into an across-part and an along-part

A tilted arrow is awkward to compute with. The trick: split it into two arrows at right angles. But "right angle to what?" — first we must nail down the axes and their positive directions.

In the figure the two black axis arrows are labelled: $+x$ across (rightward) and $+y$ downstream (upward here, drawn up only so it's visible). The plum arrow is $\vec{v}$. Drop a straight line from its tip down to the $x$-axis: the teal arrow along the axis is $v_x$, positive because it points the $+x$ way. Drop across to the $y$-axis: the orange arrow is $v_y$, positive because it points the $+y$ way. The little square marks the $90^{\circ }$ corner — the two shadows meet at a right angle, forming the right triangle whose diagonal is $\vec{v}$.

3. Adding vectors — nose to tail

The relative-velocity rule is built from adding and subtracting arrows, so we must define both.

Trace the figure with your finger: begin at the origin, walk out along the teal arrow $\vec{a}$ to its tip; from that tip walk along the orange arrow $\vec{b}$ (notice $\vec{b}$ starts where $\vec{a}$ ended). Where you finish is the tip of the plum arrow $\vec{a}+\vec{b}$, drawn straight from the origin to your final position. The plum arrow is the net trip. That is all "adding vectors" ever means.

Why "tip to tail"? Because that is literally "do the first trip, then the second trip" — the total displacement. In the river problem, "swim through the water" (trip 1) then "the water drifts over the ground" (trip 2) chains exactly like this: ${\vec{v}}_{bg}={\vec{v}}_{bw}+{\vec{v}}_w$.

That "from tip of $\vec{b}$ to tip of $\vec{a}$" picture is exactly "start at B, go to A" — the seed of the relative-velocity rule we assemble in §7.

4. Signs and the number line — 1D is just arrows on a line

Before 2D, the parent does 1D motion with plain $+$ and $-$ numbers. Here's why that's legal.

5. Reference frames & observers — what "ground" actually means

The whole topic talks about "the ground," "the water," "an observer." These are all reference frames. We must say precisely what one is before building position on top of it.

6. Position vector $\vec{r}$ and the derivative $\frac{d}{dt}$

Now that a frame gives us an origin and axes, we can locate things and watch them move.

Why this tool and not just "distance ÷ time"? Because velocity can change moment to moment; the derivative is the honest instantaneous rate — the arrow's motion at this very instant, not averaged over a whole trip. In this topic the velocities are constant, so you can safely read $\frac{d\vec{r}}{dt}$ as "the steady velocity arrow" — but the symbol is the general tool.

7. Building the subtraction rule — and why it subtracts

Now we assemble the one formula the whole topic runs on, using only §3, §5, §6.

Step 1 — WHAT is "position of $A$ as seen by $B$"? It is the arrow you'd draw from $B$ to $A$ — start where the observer $B$ sits, point to where $A$ is. By the subtraction picture of §3 (arrow from tip of ${\vec{r}}_B$ to tip of ${\vec{r}}_A$), that arrow is

${\vec{r}}_{AB}={\vec{r}}_A-{\vec{r}}_B.$

Why this and not ${\vec{r}}_A+{\vec{r}}_B$? Adding would chain two trips (§3); but "from $B$ to $A$" is a difference of positions, which is subtraction. The first index $A$ is the endpoint, the second index $B$ is the start.

Step 2 — differentiate, using the shared clock from §5. Because every frame agrees on time $t$, we may take $\frac{d}{dt}$ of both sides and it means the same $t$ on the left and right:

$\frac{d{\vec{r}}_{AB}}{dt}=\frac{d{\vec{r}}_A}{dt}-\frac{d{\vec{r}}_B}{dt}.$

Step 3 — read off velocities (§6). Each rate-of-change of a position is a velocity:

8. Angle $\theta$ and the trig ratios $\sin \theta ,\cos \theta$ — measuring the aim

The "shortest path" case needs an aiming angle. Here's the picture from scratch, with the angle convention stated explicitly.

The figure labels all three: plum hypotenuse ${\vec{v}}_{bw}$, teal horizontal leg "across $=v_{bw}\cos \theta$", orange vertical leg "upstream $=v_{bw}\sin \theta$", and the arc $\theta$ at the origin. Trace the right triangle: hypotenuse = engine speed, base = how fast you actually cross, height = how much you throw away fighting the current.

Why $\sin$ and $\cos$ and not something else? Because we split ${\vec{v}}_{bw}$ into an across-part and an upstream-part (that's §2 again). $\cos$ picks off the across leg, $\sin$ picks off the upstream leg. Setting the upstream leg equal to the current, $v_{bw}\sin \theta =v_w$, is the zero-drift condition — every symbol in it is now earned.

Since $\sin \theta \le 1$ always, $\sin \theta =v_w/v_{bw}$ is only solvable when $v_w\le v_{bw}$ — the "current can't be beaten" limit falls straight out of this picture.

How the foundations feed the topic

Equipment checklist

Connections

• Vectors — addition, components, unit vectors — the arrows, components and tip-to-tail rules built here.
• Frames of reference & Galilean transformation — why the shared clock and frame choice give ${\vec{v}}_{AB}={\vec{v}}_A-{\vec{v}}_B$.
• Newton's laws — inertial frames — where "observers" and "frames" get their physical meaning.
• Kinematics in 2D — projectile motion — same independent-perpendicular-components idea.
• Rain-man umbrella problem — reuses every symbol on this page.