Relative Motion: Quick Revision

A brief note on Relative Motion. Read it before a test. Try each question at the end before you open its answer.

The big ideas

• Relative velocity: $v_{AB} = v_A - v_B$, with signs. $v_{BA} = -v_{AB}$.
• The same way: subtract the speeds. Opposite ways: add them.
• Meeting and catching up: time $=$ gap $\div$ closing speed. Or write both positions and set them equal.
• In two dimensions the velocities add as vectors: rain slants towards a moving walker; a boat moves along the sum of its own and the river's velocity.

The facts at a glance

• Along a line: $v_{AB} = v_A - v_B$
• Km/h to m/s: multiply by $\dfrac{5}{18}$
• Catching up: $t = \dfrac{\text{gap}}{v_{\text{fast}} - v_{\text{slow}}}$
• Rain on a walker: $\tan\theta = \dfrac{v_{\text{walker}}}{v_{\text{rain}}}$
• Boat heading straight across: $t = \dfrac{d}{u}$, drift $= wt$
• Boat landing opposite: $u\sin\theta = w$, $t = \dfrac{d}{u\cos\theta}$

A worked example

A car $120\ \text{m}$ behind a bus goes at $23\ \text{m/s}$, and the bus at $15\ \text{m/s}$, the same way. When does the car reach the bus? Then a boat that moves at $3\ \text{m/s}$ in still water heads straight across a river $60\ \text{m}$ wide that flows at $1.2\ \text{m/s}$. How long does it take, and how far downstream does it land?

1. The car. $t = \dfrac{120\ \text{m}}{23\ \text{m/s} - 15\ \text{m/s}} = 15\ \text{s}$.
2. The boat's time. $t = \dfrac{60\ \text{m}}{3\ \text{m/s}} = 20\ \text{s}$.
3. The drift. $1.2\ \text{m/s} \times 20\ \text{s} = 24\ \text{m}$.

Practice with new numbers

A program to try

Mistakes to avoid

• Adding speeds of things that go the same way. Subtract them.
• Mixing km/h and m/s in one calculation.
• Mixing up A relative to B with B relative to A. They have opposite signs.
• Using the boat's speed over the ground for the crossing time. Only the part across the river carries it over.

Check yourself

1. Two trains come towards each other at $72\ \text{km/h}$ and $36\ \text{km/h}$. How fast does the gap close, in m/s?

2. A boat that moves at $4.5\ \text{m/s}$ in still water heads straight across a river $90\ \text{m}$ wide. How long does it take?

3. When does car A seem to stand still to the driver of car B?

4. The positions are $x_A = 6t$ and $x_B = 40 + 4t$, in metres, with $t$ in seconds. When and where do they meet?