JEE/NEET Physics · Motion in a Plane series · Part 4 of 6 · All parts →
- The 1-D rule generalises: v_(A rel B) = v_A − v_B, now with components
- River-boat: to land directly opposite, aim upstream at sin⁻¹(v_stream/v_boat)
- Minimum crossing time: always aim straight across (drift is the price)
- Rain-man in 2D: the umbrella tilts into the RELATIVE rain direction
- Crosswinds and airspeed vs groundspeed: pilots solve this every flight
A swimmer points straight across a river and lands downstream. A pilot points the plane north-east and still lands in Delhi on time. Both are solving the same vector puzzle: combining your motion with the world’s. Part 4 of the Motion in a Plane series.
- The 2-D subtraction rule
- Crossing rivers: two strategies
- Rain and umbrellas, upgraded
- Crosswinds: aviation’s daily puzzle
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The 2-D Subtraction Rule
Same rule as 1-D, but now component-wise: subtract the observer’s x and y parts separately. Boats, rain, wind — every 2-D relative-velocity problem is three arrows: yours, the medium’s, and the resultant that decides where you actually go.
Crossing Rivers: Two Strategies
| Goal | Aim | Result |
|---|---|---|
| Minimum TIME | straight across (⊥ to bank) | t = d/v_boat, drift = v_stream × t |
| Land directly OPPOSITE | upstream at angle sin⁻¹(v_s/v_b) | t = d/√(v_b² − v_s²), zero drift |
| v_boat ≤ v_stream | straight-across landing impossible | minimise drift: aim at cos⁻¹… upstream |
Rain and Umbrellas, Upgraded
Rain falls with vertical speed v_R; you walk at v_m. In YOUR frame the rain acquires a backward horizontal speed v_m — the rain slants toward your face, and the umbrella tilts forward by tan⁻¹(v_m/v_R). Run, and the slant steepens: everyone has felt this.
Crosswinds: Aviation’s Daily Puzzle
A plane’s AIRSPEED is its velocity relative to air; its GROUNDSPEED is relative to Earth — they differ by the wind vector. Pilots crab into crosswinds (aim upstream, like the boat) so the resultant track stays on the runway centre-line.
Solved Examples
t = 100/5 = 20 s; drift = 3 × 20 = 60 m downstream.
✔
Answer: 20 s; 60 m drift
Aim upstream at sin⁻¹(3/5) ≈ 37°.
Across-component = √(25−9) = 4 m/s → t = 100/4 = 25 s, zero drift.
✔
Answer: 37° upstream; 25 s
Relative rain = √(10² + 10²) = 14.1 m/s at 45° from vertical — the WIND slants the rain even for a standing person.
Tilt the umbrella 45° into the wind. If they also walk, subtract their velocity too — same rule, one more arrow.
✔
Answer: 45° from vertical
- Pointing the boat where you want to go. The boat goes where the VECTOR SUM points, not where its nose points — always add the stream.
- Using the boat’s full speed as the across-speed when aiming upstream. Only √(v_b² − v_s²) is crossing; the rest fights the stream.
- Adding drift and width as scalars. Drift is downstream (⊥ to width): the actual path length is Pythagoras, and the crossing time uses the across component only.
- Rain angle in the ground frame. The umbrella lives in YOUR frame — transform first, then find the angle.
This Physics in Your Daily Life
- Every pilot’s pre-flight plan computes groundspeed = airspeed + wind: a Bengaluru–Delhi flight can differ by 45 minutes each way thanks to the jet stream — same plane, different resultant.
- Ferry crossings — captains crab ferries into the current so passengers land at the exact terminal: the upstream-aim strategy, run dozens of times a day.
- Cricketers chasing a skier — the ball drifts with wind; fielders run a curve computed by their brain from the wind’s vector: 2-D relative motion at the boundary rope.
- Walking in diagonal rain — the umbrella tilt you choose IS tan⁻¹(your speed ÷ rain’s fall speed), or with wind, the full vector construction.
- Escalator and travelator walking — diagonal walks across moving walkways add your velocity to the belt’s: airports as vector laboratories.
Imagine walking across a moving walkway while it carries you sideways: to reach the shop directly opposite, you must walk diagonal — cancelling the drift with part of your stride. The stream plays the walkway; the boat plays you. Cancellation, not speed, is the goal of a straight crossing.
Boat 5, stream 3: aim 37° upstream and your 5 splits into 4 across + 3 fighting the stream — the 3 vs 3 cancels, and 4 m/s of pure crossing remains. Time: 100/4 = 25 s. Aim straight instead: full 5 across, 20 s, but 60 m of drift. Two strategies, two prices: speed vs precision.
Draw the vector triangle: boat arrow (upstream-slanted), stream arrow (downstream), resultant arrow (straight across). Slide the boat’s angle and watch the resultant swing — there is exactly ONE angle where the resultant points perfectly at the opposite bank.
Practice set (answers hidden — try first)
(NEET-level) Stream 2, boat 4, width 80 m, straight across: crossing time =
(JEE Main-level) v_b = 3, v_s = 3: can you reach the opposite point?
(NEET-level) Rain 6 down, you walk 8: relative rain speed =
(Concept) A boat’s nose points where you STEER; the boat goes where:
(JEE Main-level) Plane airspeed 200, crosswind 50 straight across: crab angle ≈
- v_rel = v_A − v_B component-wise
- straight-across aim = minimum time (drift is the price)
- upstream aim sin⁻¹(v_s/v_b) = land opposite
- needs boat > stream for zero drift
- pilots: airspeed + wind = groundspeed
- 🔁 component-wise subtraction
- 🔁 river: two strategies (time vs path)
- 🔁 straight-across landing: v_b > v_s required
- 🧠 Chant: ‘straight for speed, slant for precision’.
- 🧠 Cancellation: ‘spend 3 to fight 3, keep 4 to cross’.
- 🏠 Daily: jet-stream flight-time differences each way.
- 🏠 Daily: ferries crab into the current to dock exactly.
Quick revision
- The 1-D rule generalises: v_(A rel B) = v_A − v_B, now with components
- River-boat: to land directly opposite, aim upstream at sin⁻¹(v_stream/v_boat)
- Minimum crossing time: always aim straight across (drift is the price)
- Rain-man in 2D: the umbrella tilts into the RELATIVE rain direction
- Crosswinds and airspeed vs groundspeed: pilots solve this every flight
- Crossing rivers: two strategies
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