Change your seat
Car A starts at +10 m/s and car B at −20 m/s. The sign post C stands still. Pick where you sit and watch how the same road looks different.
Positive means to the right, negative means to the left. The yellow outline marks the vehicle you are sitting in. Notice that the road itself and the sign C also seem to move when you sit in a moving car.
The rule — 20.1 same direction · 20.2 opposite direction
To find the velocity of X relative to Y, subtract the velocity of Y, because you are removing your own motion from what you see.
Opposite directions
Give one of them a minus sign. Subtracting a negative number adds, so the relative speed is the sum of the two speeds: 10 + 20 = 30 m/s.
Same direction
Both have the same sign, so the relative speed is the difference of the two speeds: 30 − 20 = 10 m/s.
Swap the viewer
The answer keeps its size and flips its sign: VAB = −VBA.
Here is the example from your notes: A = +10 m/s, B = −20 m/s, C stationary. Each cell is the velocity of the row object as seen from the column object.
| Object | Seen from A | Seen from B | Seen from C (ground) |
|---|---|---|---|
| A | 0 | +30 m/s | +10 m/s |
| B | −30 m/s | 0 | −20 m/s |
| C | −10 m/s | +20 m/s | 0 |
Look at the middle column of the top row: VAB = 10 − (−20) = +30 m/s. That is the same +30 you saw when you sat in B.
The four-step method
- Choose the positive direction. Usually the direction of the first object.
- Write every velocity with its sign. Anything moving the other way gets a minus.
- Subtract the viewer. For X relative to Y, calculate VX − VY.
- Read the answer. The sign tells the direction, the size tells the speed. Check the units.
Mistakes to avoid:
- Forgetting to convert km/h to m/s. Multiply by 5/18, so 72 km/h = 20 m/s.
- Subtracting in the wrong order. The viewer's velocity is always the one taken away.
- Losing the minus sign of the opposite-moving object.
- In meeting and crossing problems, dividing by one speed instead of the relative speed.
20.3 — At an angle (the general case)
When two velocities are neither parallel nor opposite, "add a minus sign" is not enough. Here is the trick: draw both velocities as arrows starting from the same point, to the same scale. The arrow that joins their two tips — pointing toward the object you want the velocity of — is the relative velocity. No reversing, no reassembling: just join the tips.
You may also see this written as R = √(v1² + v2² + 2v1v2cos α), where α is the angle after reversing the viewer's vector first. Both give the same number: α is simply the supplement of β (α = 180° − β), and cos(180° − β) = −cos β, so the two formulas are the same formula written two ways. Use whichever picture is easier to draw.
Blue: A. Orange: B. Green: the relative velocity you asked for, drawn tip-to-tip. Try β = 0° and β = 180° — you should get back the same-direction and opposite-direction rules from above.
21. River–boat problems
A boat with still-water speed Vb crosses a river of width d that flows at Vt. Whatever angle the boat is steered at, its actual path over the ground is the vector sum of these two velocities. There are three classic ways to steer, and they trade off against each other.
Type 1 — shortest distance
Steer upstream so the current is exactly cancelled: zero drift, but the longest crossing time. Needs Vb > Vt.
Type 2 — shortest time
Steer straight across (90° to the bank). This is the fastest crossing, t = d / Vb, but the current still carries you downstream.
Type 3 — with the flow
Steer downstream on purpose. Both drift and time increase — useful only when a specific downstream point is the real target.
Let θ be the angle the boat's heading makes with the straight-across direction: positive θ means aimed upstream, negative θ means aimed downstream.
Positive θ = aimed upstream (fights the current). Negative θ = aimed downstream ("with the flow"). The diagram is schematic, not drawn exactly to scale for extreme values.
22. Rain problems
Find the velocity of the rain relative to the observer — that is the direction the rain seems to come from, and it is exactly the direction an umbrella should be tilted along. Rain always falls straight down relative to the ground; wind and walking only change what it looks like to the person underneath it.
Type 1
Rain falls, pedestrian moves, no wind. Just subtract the pedestrian's velocity from the rain's.
Type 2
Rain falls, wind blows, pedestrian stands still. The wind alone tilts the rain.
Type 3
Wind and pedestrian move the same way. Combine wind with rain first, then subtract the pedestrian.
Type 4
Wind blows against the pedestrian. Same two steps, opposite sign on the wind.
Type 5
An inclined road, or extra axes. Same method — resolve everything onto a common set of axes first.
Blue streaks: the rain's velocity relative to you. Orange: the umbrella, held along the streaks. Turn on a wind to see Types 2–4.
Check yourself #check-yourself
Stage 4 of the loop. Six quick questions across the whole topic — answer in any order, retry a wrong one straight away, and read the reason every time.
0 of 6 correct