Physics · Mechanics · Grade 7

Vectors and relative motion

When a boat crosses the Amu Darya, why does it land downstream rather than directly opposite?

The situation

You are in a motorboat and want to cross the Amu Darya from a pier on one bank to the pier directly opposite on the other. The river flows strongly: if you point the boat's bow straight at the opposite bank, the current carries it downstream. A person standing on the bank, a fisherman on a raft drifting with the current, and you in the boat all see the same motion differently. Which way should the boat be pointed, and which route is the fastest?

Topic

Vectors and the relativity of motion. Crossing the Amu Darya by boat: boat and current velocities add up

Key ideas

Addition of vectors
Addition of velocities
Galilean relativity

Formulas

\vec{V} = \vec{v} + \vec{u}

The classical law of addition of velocities: velocity relative to the bank = velocity relative to the water + velocity of the current.

x^{'} = x - u t, y^{'} = y, t^{'} = t

Galilean transformations: changing to the frame of reference that moves with the current (the raft).

V_{x} = u - v \sin α, V_{y} = v \cos α

The vector equation in terms of its components along the coordinate axes (scalar form); $α$ is the angle between the boat's bow and the perpendicular to the bank.

t = \frac{d}{v \cos α}, s = (u - v \sin α) t, \sin α^{*} = \frac{u}{v}

The crossing time, the drift along the current, and the condition for crossing without drift ( $v > u$ ).

\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos φ = a_{x} b_{x} + a_{y} b_{y}

The scalar product of two vectors is a number (a scalar).

Theory

Scalar and vector quantities

Mass, time, distance and energy are fully given by a single number: these are scalar quantities. Displacement, velocity and force also need a direction — these are vectors. A vector can be specified in two equivalent ways: by its magnitude and direction ( $| \vec{a} |$ , $θ$ ) or by its components ( $a_{x} = | \vec{a} | \cos θ$ , $a_{y} = | \vec{a} | \sin θ$ , $| \vec{a} | = \sqrt{a_{x}^{2} + a_{y}^{2}}$ ). In “Vectors” mode the sliders change the first way and dragging with the mouse changes the second — the result is the same. The position of a body is given by the position vector $\vec{r}$ drawn to it from the origin; velocity is its derivative $\vec{v} = d \vec{r} / d t$ , and displacement is the integral of velocity.

Addition, subtraction and the scalar product

Vectors are added by the triangle rule (the second is placed at the tip of the first) or the parallelogram rule. The difference $\vec{a} - \vec{b} = \vec{a} + (- \vec{b})$ is the vector pointing from the tip of $\vec{b}$ to the tip of $\vec{a}$ . The scalar product $\vec{a} \cdot \vec{b} = | \vec{a} | | \vec{b} | \cos φ$ is the projection of $\vec{a}$ onto the direction of $\vec{b}$ × $| \vec{b} |$ . For perpendicular vectors it is zero, and then $| \vec{a} + \vec{b} | = | \vec{a} - \vec{b} |$ , because $| \vec{a} \pm \vec{b} |^{2} = a^{2} + b^{2} \pm 2 \vec{a} \cdot \vec{b}$ . A physical formula can be written in vector form ( $\vec{V} = \vec{v} + \vec{u}$ , $m \vec{a} = \vec{F}$ ) or as scalar equations along the axes — the meaning is the same.

Combining motions and the principle of independence

The boat's compound motion relative to the bank is the sum of two independent simple motions: with velocity $\vec{v}$ relative to the water, and together with the water at velocity $\vec{u}$ . They do not interfere with each other: the crossing time depends only on the transverse component $V_{y} = v \cos α$ , while the current only produces drift. So the fastest route is $α = 0$ ( $t_{m i n} = d / v$ ). To cross without drift, the bow must be turned upstream by an angle with $\sin α = u / v$ — this is possible only if $v > u$ and takes longer. If $v < u$ , the least drift occurs at $\sin α = v / u$ : $s_{m i n} = d \sqrt{u^{2} - v^{2}} / v$ .

Frames of reference and Galilean relativity

Motion is always described relative to some frame of reference (a body + coordinates + a clock). From the bank the boat's track is a slanting straight line; from the raft it is a straight line along the bow; and from the boat the bank and the raft sail away backwards. The trajectory and the speed are relative, but time, acceleration and the duration of the crossing are the same in every frame. An inertial frame is one in which a free body moves uniformly in a straight line (Newton's first law); any frame moving uniformly relative to it is also inertial. Galileo's principle: the laws of mechanics are the same in all inertial frames — the choice of frame does not change the laws. When the boat pulls away or turns, it becomes a non-inertial frame: the passenger is pushed by inertial forces (see the experiments “Foucault” and “Newton's laws”).

Limits of validity

Galileo's law $\vec{V} = \vec{v} + \vec{u}$ holds when the speeds are much smaller than the speed of light $c$ . At speeds close to it, the Lorentz transformations and the relativistic addition law $V = (v + u) / (1 + v u / c^{2})$ are used (the “Special relativity” experiment). For the boat the correction $v u / c^{2} \sim 10^{- 16}$ — no instrument would detect it. In theoretical mechanics Galilean invariance looks like this: the Lagrangian of a free particle $L = m v^{2} / 2$ changes under a change of frame only by a total time derivative, while the equations of motion do not change. The model is simplified: the current is the same across the whole width (in reality it is slower near the bank), and the boat's speed is established instantly.

Adding forces and resolving them into components

A barge is pulled by two tugs. The resultant of forces applied at one point is found by the parallelogram rule: $\vec{R} = {\vec{F}}_{1} + {\vec{F}}_{2}$ , $R_{x} = F_{1} \cos θ_{1} + F_{2} \cos θ_{2}$ . If the forces are applied at different points of the body, we slide them along their lines of action to the point where they intersect and add them there — $\vec{R}$ passes through that point. For parallel forces $R = F_{1} + F_{2}$ , and the point of application divides the gap in inverse proportion to the forces: $F_{1} a_{1} = F_{2} a_{2}$ . If $\vec{R}$ does not pass through the centre of mass, the moment of the force turns the barge. Each force can be resolved into components: along the barge's axis (pulling forwards) and perpendicular to it (pushing sideways); if the sideways components cancel out, the barge goes straight.

Did you know?

Ancient Greek and Roman authors called the Amu Darya the Oxus, and in medieval Arabic and Persian sources it is mentioned under the name Jayhun.
In 1632, in his “Dialogue Concerning the Two Chief World Systems”, Galileo described a thought experiment in a ship's cabin: if the ship sails smoothly, the butterflies, fish and drops inside behave just as they do on shore — no experiment can reveal the ship's motion.

Inside the experiment

The experiment starts from a real-life situation: you predict the result, change the controls, run the experiment and draw a conclusion. Inside are missions, theory, a quiz and a guided tour in which the camera moves and explains the key points.

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