Relative velocity in one dimension

The car that stands still

On a highway, a car beside yours, at the same speed, seems to stand still. A car that overtakes you creeps past slowly. A car on the other side of the road flashes by. All three move at about the same speed over the road. What changes is how they move compared with you.

The relative velocity

Along one line, the velocity of A as seen from B is

It is what B would measure if B thought itself at rest. Two cases, with speeds $v_A$ and $v_B$:

• Same direction: $v_{AB} = v_A - v_B$, the difference of the speeds.
• Opposite directions: with B's velocity negative, $v_{AB} = v_A - (-v_B) = v_A + v_B$, the sum.

And $v_{BA} = -v_{AB}$: B sees A move one way, and A sees B move the other way at the same speed.

Watch the gap

The gap between the cars changes at the rate $v_A - v_B$. When the lines cross, the gap is zero: A has caught up.

Worked examples

Catching a bus. A scooter at $12\ \text{m/s}$ chases a bus at $8\ \text{m/s}$ that is $100\ \text{m}$ ahead.

1. Sit on the bus. The scooter comes closer at $12\ \text{m/s} - 8\ \text{m/s} = 4\ \text{m/s}$.
2. Time. $\dfrac{100\ \text{m}}{4\ \text{m/s}} = 25\ \text{s}$. (Without the relative-velocity idea, you would solve $12\ \text{m/s}\,t = 100\ \text{m} + 8\ \text{m/s}\,t$ and get the same.)

Two trains passing. A train $90\ \text{m}$ long at $15\ \text{m/s}$ meets one $110\ \text{m}$ long at $25\ \text{m/s}$, coming the other way on the next track.

1. Relative speed. $15\ \text{m/s} + 25\ \text{m/s} = 40\ \text{m/s}$.
2. Distance. From fronts meeting to backs parting, the two lengths: $200\ \text{m}$.
3. Time. $\dfrac{200\ \text{m}}{40\ \text{m/s}} = 5\ \text{s}$.

Watch out. For overtaking and passing, the distance to gain is both lengths, not one. And check the directions before you add or subtract speeds.

Practise with new numbers

Try it in Python

The program finds when the scooter catches the bus, and checks it second by second.

Remember these

• $v_{AB} = v_A - v_B$, with signs. Same way: subtract the speeds. Towards each other: add them.
• Sit on one body; the other moves at the relative velocity.
• Overtaking and passing: the distance to gain is both lengths.

Check yourself

1. Car A moves east at $22\ \text{m/s}$ and car B east at $16\ \text{m/s}$. Find the velocity of A relative to B, and of B relative to A.

2. Two runners on a straight path start $150\ \text{m}$ apart and run towards each other at $4\ \text{m/s}$ and $6\ \text{m/s}$. When do they meet?

3. Rain falls straight down. To a passenger in a moving train, it seems to slant. Which way, and why?

4. A boy on a train walks towards the front at $1.2\ \text{m/s}$. The train moves at $20\ \text{m/s}$. How fast does he move as seen from the platform?