Velocity and Relative Motion in a Plane: Quick Revision

A brief note on Velocity and Relative Motion in a Plane: how a body moves in a plane, and how its motion looks from another moving body. Try each question at the end before you open its answer.

The big ideas

• The position vector $\vec{r} = x\hat{i} + y\hat{j}$ says where a body is, from a chosen origin.
• The displacement is the change of position: $\Delta\vec{r} = \vec{r}_2 - \vec{r}_1$.
• The average velocity is $\vec{v}_{\text{av}} = \Delta\vec{r}/\Delta t$. It points along the displacement.
• The velocity at an instant is the average velocity over a very short time. It always points along the path.
• The acceleration is $\vec{a} = \Delta\vec{v}/\Delta t$. A change of direction is a change of velocity, so a body that turns at a steady speed is accelerating.

Steady acceleration: two motions at once

With a constant acceleration, the $x$ motion and the $y$ motion run side by side. Each has its own equations:

and the same for $y$. As vectors, from the origin: $\vec{v} = \vec{u} + \vec{a}t$ and $\vec{r} = \vec{u}t + \tfrac{1}{2}\vec{a}t^2$.

An example. A toy hovercraft glides east across a smooth floor at $4\ \text{m/s}$. Its fan is turned to push it north, with a steady acceleration of $2\ \text{m/s}^2$. After $3\ \text{s}$:

• $\vec{v} = 4\ \text{m/s}\,\hat{i} + 2\ \text{m/s}^2 \times 3\ \text{s}\,\hat{j} = (4\hat{i} + 6\hat{j})\ \text{m/s}$, so its speed is $\sqrt{16\ \text{m}^2/\text{s}^2 + 36\ \text{m}^2/\text{s}^2} = 7.21\ \text{m/s}$.
• $\vec{r} = 4\ \text{m/s} \times 3\ \text{s}\,\hat{i} + \tfrac{1}{2} \times 2\ \text{m/s}^2 \times (3\ \text{s})^2\,\hat{j} = (12\hat{i} + 9\hat{j})\ \text{m}$, so it is $\sqrt{(12\ \text{m})^2 + (9\ \text{m})^2} = 15\ \text{m}$ from the start.

Relative velocity

The velocity of $A$ as seen from $B$ is

It is what $B$ measures while thinking itself at rest.

• Quickest: head: straight across; time: $t = d/v_b$
• Straight opposite: head: upstream, $\sin\theta = v_r/v_b$; time: $t = d/\sqrt{v_b^2 - v_r^2}$

Here $d$ is the river's width, $v_b$ the boat's speed in still water and $v_r$ the river's speed. On the quickest crossing the boat drifts $v_r t$ downstream.

Rain. Rain falling straight down seems, to a person walking, to slant towards them. So the umbrella tilts forward, at $\tan\theta = v_{\text{walk}}/v_{\text{rain}}$ from the vertical.

An example: a bird in a crosswind. A bird flies at $15\ \text{m/s}$ relative to the air, pointing north. A wind blows from west to east at $4\ \text{m/s}$.

• Over the ground it moves at $\sqrt{(15\ \text{m/s})^2 + (4\ \text{m/s})^2} = 15.5\ \text{m/s}$, at $\tan^{-1}(4\ \text{m/s}/15\ \text{m/s}) = 14.9^\circ$ east of north.
• To travel due north, it must point into the wind, at $\sin\theta = 4\ \text{m/s}/15\ \text{m/s}$, which is $15.5^\circ$ west of north. Its ground speed is then $\sqrt{(15\ \text{m/s})^2 - (4\ \text{m/s})^2} = 14.5\ \text{m/s}$.

Practice with new numbers

A program to try

Two ships at sea: how does each one's motion look from the other?

Mistakes to avoid

• Average speed and average velocity mixed up. On a curved path the distance is longer than the displacement, so the average speed is larger than the size of the average velocity.
• The order in relative velocity. $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$. Swap them and the answer turns round.
• The two river questions. "The quickest crossing" and "straight across" need different headings. Read which one is asked.
• A steady speed means no acceleration. Not in a plane: a body going round a bend at a steady speed is accelerating.

Check yourself

1. A runner goes once round a circular track of radius $35\ \text{m}$ in $50\ \text{s}$. What is the average speed for the lap, and the size of the average velocity?

2. Can a body have zero velocity and still be accelerating? Give an example.

3. A particle moves with $\vec{v} = (8\hat{i} + 15\hat{j})\ \text{m/s}$. What is its speed, and at what angle to the $x$ axis does it move?

4. Two cars approach a crossing. Car $A$ drives north at $12\ \text{m/s}$, and car $B$ drives east at $9\ \text{m/s}$. How fast does $A$ move, as seen by the driver of $B$?

5. A swimmer swims at $1.2\ \text{m/s}$ in still water. A river flows at $1.8\ \text{m/s}$. Can the swimmer land straight opposite the starting point?