JEE/NEET Physics · Motion in a Straight Line series · Part 5 of 6 · All parts →
- Velocity is always measured relative to something — there is no absolute ‘truly at rest’
- v_A relative to B = v_A − v_B (vector subtraction: subtract the observer)
- Same direction: speeds subtract; opposite: add; perpendicular: Pythagoras
- River-boat: across-time uses the across-component; drift uses the stream; to cross straight, aim upstream
- Rain-man: tilt your umbrella into the rain’s velocity RELATIVE to you
Sit in a moving train and watch another train slide past backwards — is it moving, or are you? Both answers are right; it depends on the observer. That’s relative velocity. Part 5 of the Motion in a Straight Line series.
- No absolute rest
- The one rule: subtract the observer
- Rain, rivers, and trains
- Crossing a river the smart way
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
No Absolute Rest
You are ‘sitting still’ — on a planet spinning at 1600 km/h, orbiting the Sun at 30 km/s, racing around the galaxy at 220 km/s. Every velocity ever quoted is relative to some observer. Physics takes this seriously: pick your observer, then measure.
The One Rule: Subtract the Observer
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| v_A | velocity of A, measured from ground | m/s |
| v_B | velocity of the observer B | m/s |
| v_(A rel B) | how A looks from B’s window | m/s |
Rain, Rivers, and Trains
| Setup | Relative velocity | Everyday meaning |
|---|---|---|
| Two cars, same direction 60 & 40 | 60 − 40 = 20 | overtaking feels slow |
| Two cars, opposite 60 & 40 | 60 + 40 = 100 | passing feels violent |
| Rain 10 down, man walks 5 | √(10²+5²) ≈ 11.2 slanted | umbrella tilts forward |
| Boat 3 across, stream 4 | resultant 5 at an angle | drift downstream |
Crossing a River the Smart Way
Boat speed b in still water, stream s, river width d. Minimum time: point straight across — t = d/b, but you drift s·t downstream. Shortest path (land exactly opposite): aim upstream so your across-component is √(b² − s²) — possible only if b > s.
Solved Examples
Same direction subtracts: 20 − 15 = 5 m/s — the slow crawl you feel when overtaking.
Opposite would be 35 m/s. ✔
Answer: 5 m/s
Rain relative to you: backward 5, down 10 → angle from vertical = tan⁻¹(5/10).
≈ 26.6° tilted forward — your motion makes the rain accuse you of walking into it. ✔
Answer: tan⁻¹(1/2) ≈ 26.6° forward
Boat (3) < stream (4): even aiming fully upstream, you can’t cancel the drift.
No — minimum drift is 60 × 4/√(3²+4²)… careful: aim 53° upstream, drift = 60×(4−3cos53°…)/… The core point: b must exceed s to cross straight.
Best drift: aim at cosθ = 3/4 upstream → net downstream speed = 4 − 3(3/4) = 1.75, drift = 60 × 1.75/2.4 ≈ 43.8 m. ✔
Answer: No — boat slower than stream; min drift ≈ 44 m
- Adding instead of subtracting. Same-direction relative motion SUBTRACTS; only opposite directions add.
- Forgetting it’s a vector subtraction. Perpendicular velocities combine by Pythagoras, not plain addition.
- Minimum-time vs minimum-path confusion. Straight-across aim gives minimum time WITH drift; reaching the opposite point needs upstream aim (and a faster boat).
- Solving rain problems in the ground frame. Always transform to the walker’s frame first — the umbrella lives there.
This Physics in Your Daily Life
- Overtaking on highways feels gentle at 5 m/s relative but head-on passing at 35 m/s relative is terrifying — your brain is a relative-velocity computer deciding when it’s safe to pass.
- Walking in rain with an umbrella tilted forward — the tilt angle IS tan⁻¹(your speed ÷ rain’s fall speed), computed instinctively by millions of commuters daily.
- Airplane ‘groundspeed vs airspeed’ — a 900 km/h plane in a 100 km/h tailwind crosses the ground at 1000: flight times east vs west differ by hours on the same route.
- Escalator walking — walking up a moving escalator adds your speed to the stair speed: the everyday vector sum, felt as arriving sooner.
- Catching a ball from a moving car — your brain subtracts the car’s velocity to judge the catch: relativity at the playground.
Two prisoners in a windowless smooth train can’t tell if they’re moving — only by looking OUT at the ground does motion become a statement. Velocity is never a property of one object alone; it’s a property of a PAIR: object and observer. Asking ‘how fast is it REALLY moving?’ is like asking ‘is it far?’ without saying far from what.
You at 5 m/s, cyclist at 8 same direction: he recedes at 3 m/s — gentle. Oncoming at 8: he approaches at 13 — alarming. The runners on adjacent tracks of a stadium move at many velocities at once: fast relative to oncoming runners, zero relative to same-speed pacers. One runner, infinitely many true speeds.
Draw arrows tail-to-tail: your velocity arrow, the observer’s arrow; the RELATIVE velocity is the arrow from the observer’s tip to yours (the subtraction drawn). Same-direction arrows leave a short gap; opposite arrows span a long bridge. Every relative-velocity problem is this one picture.
Practice set (answers hidden — try first)
(NEET-level) Cars 72 and 54 km/h opposite: relative speed =
(JEE Main-level) Rain 10 down, wind 5 horizontal: rain’s speed relative to ground =
(NEET-level) Boat 5 across, stream 3 (perp): resultant =
(Concept) Two cars at equal speed side by side: relative velocity =
(JEE Main-level) 120 m river, boat 4, stream 3, straight-across aim: crossing time =
- v_rel(A,B) = v_A − v_B (vectors)
- same direction subtract, opposite add, perpendicular → Pythagoras
- rain-man: tilt = tan⁻¹(v_walk/v_rain)
- min time: aim straight across (drift unavoidable)
- straight-across landing needs boat > stream
- 🔁 relative velocity = subtract observer’s velocity
- 🔁 trains: same −, opposite +
- 🔁 rain-man umbrella angle
- 🧠 Chant: ‘subtract the seer’.
- 🧠 Highway feel: ‘overtake subtracts, head-on adds’.
- 🏠 Daily: umbrella tilt = your speed vs rain’s fall.
- 🏠 Daily: east-west flights differ by jetstream hours.
Quick revision
- Velocity is always measured relative to something — there is no absolute ‘truly at rest’
- v_A relative to B = v_A − v_B (vector subtraction: subtract the observer)
- Same direction: speeds subtract; opposite: add; perpendicular: Pythagoras
- River-boat: across-time uses the across-component; drift uses the stream; to cross straight, aim upstream
- Rain-man: tilt your umbrella into the rain’s velocity RELATIVE to you
- The one rule: subtract the observer
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