NEET Physics — Motion In One Dimension
Relative Velocity – Complete Notes, Revision, Important Questions & Downloads
Relative velocity describes the motion of one body as observed from another moving body using the vector rule v₁₂ = v₁ − v₂. Five subtopics are covered: Introduction and General Formula, Relative velocity of Satellite, Relative velocity of Rain, Relative velocity of Swimmer, and Crossing the River. NEET tests this topic through calculation questions — typical format gives two objects with velocities in specified directions and asks for magnitude or direction of relative velocity. The crossing-the-river problem, with its shortest-distance (cos θ = vr/vm) and shortest-time (swim perpendicular) strategies, appears in NEET every 2–3 years as a numerical.
9 SubtopicsVector Subtractionv₁₂ = v₁ − v₂
Expected QuestionsQ
1
Relative velocity contributes 1 question per NEET paper on average, either as a stand-alone velocity-difference calculation, a rain-umbrella angle problem, or a river-crossing time/distance question.
Time Required⏱
2 hrs
45 min for general formula and direction cases; 45 min for rain/swimmer/satellite applications; 30 min for both river-crossing strategies + 3–4 numerical drills.
Difficulty⚡
Medium
The algebra of vector subtraction is straightforward, but direction analysis for river crossing at minimum distance (angle with upstream, not perpendicular) is a persistent source of error.
NRI USA Curriculum GapUS
Medium
US AP Physics covers relative motion in 1D but rarely drills the river-crossing minimum-distance strategy (cos θ = vr/vm) or rain-relative-velocity umbrella-angle problems, which are NEET staples.
9Subtopics
10+Practice Questions
4Free Downloads
2 hrsPrep Time
NEET Weightage — Relative Velocity
Motion In One Dimension (Chapter 2)| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2024 | 1 | 4 | |
| 2023 | 0 | 0 | |
| 2022 | 1 | 4 | |
| 2021 | 1 | 4 | |
| 2020 | 0 | 0 | |
| 2019 | 1 | 4 | |
| 6-Year Total (2019–2024) | 3–5 | 12–20 |
Same-direction formula: v₁₂ = v₁ − v₂. Opposite direction: v₁₂ = v₁ + v₂. These two cases are the most tested on NEET — confusing them (adding instead of subtracting) is the most common error.
River crossing — shortest distance: swim at angle θ with upstream where cos θ = vr/vm; time = w/√(vm² − vr²). Shortest time: swim perpendicular to bank; time = w/vm; drift = (vr/vm)·w.
Rain-relative-velocity angle θ = tan⁻¹(vM/vR) with vertical. NEET often gives the angle and asks for speed of rain or observer — set up the right triangle correctly before solving.
River crossing — shortest distance: swim at angle θ with upstream where cos θ = vr/vm; time = w/√(vm² − vr²). Shortest time: swim perpendicular to bank; time = w/vm; drift = (vr/vm)·w.
Rain-relative-velocity angle θ = tan⁻¹(vM/vR) with vertical. NEET often gives the angle and asks for speed of rain or observer — set up the right triangle correctly before solving.
0.7
Avg Questions / Year
16
Total Marks (6 yrs)
Mixed
Pattern
Medium
Difficulty
How to Prepare Relative Velocity for NEET
1
Nail the direction rules first Write v₁₂ = v₁ − v₂ for general case. Memorise: same direction → subtract magnitudes; opposite → add. Drill 4–5 examples distinguishing these two before moving to 2D cases.
2
Practice rain problems with a right-triangle sketch Always draw: vertical arrow for rain (vR), horizontal arrow for observer (vM), resultant for rain-relative-observer. The angle with vertical is tan⁻¹(vM/vR). Never guess from the diagram — compute it.
3
Master both river-crossing strategies separately Shortest distance: swimmer angles against current; resultant velocity is perpendicular to bank. Shortest time: swimmer goes straight across; drift appears downstream. Confusing the two strategies is a 4-mark trap.
4
Satellite relative velocity — one formula, two cases West-to-east (same as Earth rotation): vse = vs − ve. East-to-west (opposite Earth rotation): vse = vs + ve. NEET asks which direction gives higher relative speed — the answer is always east-to-west.
Study Materials — Relative Velocity
PDF · Cheat Sheet · MCQ Set · PYQFull Notes
Complete theory for all 5 subtopics: general formula derivation, all direction cases, satellite/rain/swimmer/river-crossing worked examples with formula chains.
Download NotesFormula Sheet
One-page reference with all relative velocity formulas: v₁₂ = v₁ − v₂, rain angle, swimmer resultant, river-crossing shortest-distance and shortest-time expressions.
Download SheetMCQ Practice
25 scenario-based problems covering all subtopics: direction-based relative velocity, rain angle computation, swimmer velocity components, and river-crossing optimisation.
Download MCQsPYQ
Curated NEET-style previous-year questions on relative velocity — velocity calculation, rain-umbrella angle, river width and drift problems with step-by-step solutions.
Download PYQsSubtopics in Relative Velocity
2-Column Table| Column A | Column B |
|---|---|
Rapid Revision — Relative Velocity
Concept → Trap → Example1) Introduction and General Formula
Core Formulav₁₂ = v₁ − v₂ (vector subtraction). Same direction: |v₁₂| = |v₁ − v₂|. Opposite: |v₁₂| = v₁ + v₂. At angle θ: |v₁₂| = √(v₁² + v₂² − 2v₁v₂cosθ).
- Relative velocity is always v₁ − v₂, not v₂ − v₁. The subscript order matters: v₁₂ means velocity of 1 relative to 2.
- Perpendicular case: v₁₂ = √(v₁² + v₂²). Useful for rain problems where vR and vM are at 90°.
- Trap: When two cars travel in the same direction at 40 and 60 km/h, relative speed = 20 km/h, not 100 km/h — do NOT add them.
Example (NEET-style)Car A at 80 km/h east, car B at 50 km/h east: vAB = 80 − 50 = 30 km/h east. If B travels west: vAB = 80 + 50 = 130 km/h east.
2) Relative velocity of Satellite
ApplicationSatellite velocity relative to Earth's surface: vse = vs − ve (west-to-east orbit). Opposite orbit: vse = vs + ve (east-to-west).
- Earth rotates west-to-east at ~0.46 km/s at equator. A geostationary satellite moves at the same angular speed — relative velocity = 0.
- Satellite in opposite direction (retrograde orbit): relative velocity = vs + ve, making it appear to move faster across the sky.
- NEET asks: which satellite has higher relative velocity to an observer on Earth? Answer — retrograde (east-to-west) orbit.
Example (NEET-style)Satellite at vs = 7.9 km/s (west-to-east), ve = 0.46 km/s: vse = 7.9 − 0.46 = 7.44 km/s. Retrograde orbit: 7.9 + 0.46 = 8.36 km/s.
3) Relative velocity of Rain
GeometryvRM = vR − vM. Magnitude: vRM = √(vR² + vM²). Direction with vertical: θ = tan⁻¹(vM/vR). Person must tilt umbrella at angle θ forward.
- Rain falls vertically (vR downward); observer moves horizontally (vM forward). Relative velocity of rain has both downward and backward components relative to observer.
- The umbrella must be tilted forward (in direction of motion) at angle θ with vertical, where tanθ = vM/vR.
- Trap: NEET may give θ and ask for vM. Use vM = vR tanθ — don't flip the ratio.
Example (NEET-style)Rain at vR = 20 m/s (vertical), person at vM = 20 m/s: vRM = √(400+400) = 20√2 m/s. θ = tan⁻¹(20/20) = 45° — tilt umbrella at 45°.
4) Relative velocity of Swimmer
Vector AdditionSwimmer velocity relative to ground: vM = v + vR (vector). With flow: |vM| = v + vR. Against flow: |vM| = v − vR.
- The swimmer's velocity v is relative to water, not the ground. Ground velocity = swimmer velocity + river velocity (vector sum).
- Swimming in flow direction boosts ground speed; swimming against flow reduces it. Net speed must be positive (v > vR) or swimmer drifts backward.
- Trap: confusing the reference frame — 'v can swim at 3 m/s' means relative to water, not ground.
Example (NEET-style)Swimmer at v = 5 m/s, river at vR = 3 m/s. With flow: 5 + 3 = 8 m/s. Against: 5 − 3 = 2 m/s. Zero if vR > v — swimmer cannot go upstream.
5) Crossing the River
OptimisationShortest distance: swim at angle θ upstream where cosθ = vr/vm; time t₁ = w/√(vm² − vr²). Shortest time: swim perpendicular; t₂ = w/vm; drift = (vr/vm)·w.
- Minimum distance (straight-line cross): the resultant velocity must be perpendicular to banks. Swimmer aims upstream by angle θ = cos⁻¹(vr/vm). Condition: vm > vr.
- Minimum time: aim perpendicular to banks regardless of drift. This gives shortest crossing time t₂ = w/vm. The downstream drift of vr·(w/vm) is accepted.
- Trap: Many students use sinθ = vr/vm for minimum distance — the correct relation is cosθ = vr/vm (angle with upstream, not with the bank).
Example (NEET-style)River width w = 100 m, vm = 5 m/s, vr = 3 m/s. min-time: t = 100/5 = 20 s, drift = (3/5)×100 = 60 m. min-distance: t = 100/√(25−9) = 100/4 = 25 s, drift = 0.
US Curriculum Gaps — Relative Velocity
Topics in this section are tested in NEET but covered less rigorously in standard US physics courses.River-Crossing Optimisation (AP Physics 1 Gap)
AP Physics 1 introduces relative motion conceptually but does not include the organised treatment of river-crossing with two separate optimisation strategies (minimum distance vs minimum time). NEET requires students to derive the angle condition cosθ = vr/vm and use it in numerical problems.
- AP Physics 1 covers relative velocity in 1D and 2D but not river-crossing geometry as a distinct NEET subtopic
- The formula t₁ = w/√(vm²−vr²) for shortest-distance crossing is not in AP Physics 1 curriculum
- Drift computation AB = (vr/vm)·w for minimum-time crossing requires specific drill not in US textbooks
Rain and Umbrella Angle Problems (AP Physics C Mechanics Gap)
AP Physics C: Mechanics covers relative velocity vectors but the applied scenario of rain-relative-observer angle (θ = tan⁻¹(vM/vR) with vertical) and umbrella tilting direction is a standard NEET question type absent from the AP C exam. US students would need additional practice with this specific applied topic.
- Rain-relative-velocity problems testing direction of umbrella tilt are NEET-specific application questions
- The geometric relationship between rain vertical component and observer horizontal speed is not a dedicated AP Physics C topic
- NEET frequently tests: given θ and vR, find vM — requiring reversal of the tan formula
NEET-Style Practice Questions — Relative Velocity
5 Questions1A boat can row at 4 m/s in still water. The river flows at 3 m/s. If the boat rows perpendicular to the bank, what is the magnitude of the net velocity of the boat?Swimmer Relative Velocity
5 m/s
7 m/s
1 m/s
4 m/s
When the boat rows perpendicular to the bank, its velocity relative to ground is the vector sum of boat velocity (4 m/s, perpendicular to bank) and river velocity (3 m/s, along bank). Since these are perpendicular: |vM| = √(4² + 3²) = √(16 + 9) = √25 = 5 m/s. Option B (7 m/s) is wrong — you cannot add magnitudes of perpendicular vectors. Option C (1 m/s) would be correct only if the velocities opposed each other. Option D ignores the river.
2Rain falls vertically at 10 m/s. A man walks at 10 m/s horizontally. At what angle with the vertical should he hold his umbrella?Relative velocity of Rain
30°
45°
60°
90°
Velocity of rain relative to man: vRM = vR − vM. Since vR = 10 m/s (downward) and vM = 10 m/s (horizontal), these are perpendicular. The angle with vertical: tanθ = vM/vR = 10/10 = 1, so θ = tan⁻¹(1) = 45°. The umbrella must be tilted 45° forward (in direction of motion). 30° underestimates the horizontal component; 60° overestimates it. 90° would mean holding umbrella horizontally, which is not protective.
3Two trains move in opposite directions at 60 km/h and 40 km/h. What is the relative velocity of the first train with respect to the second?Introduction and General Formula
20 km/h
100 km/h
50 km/h
60 km/h
For objects moving in opposite directions, relative velocity = v₁ + v₂ (magnitudes are added because the velocity vectors point in opposite directions, and subtraction of a negative vector gives addition). v₁₂ = 60 − (−40) = 100 km/h. Option A (20 km/h) is the relative speed for same-direction motion. Option C and D are incorrect. The general formula v₁₂ = v₁ − v₂ gives 60 − (−40) = 100 km/h when v₂ is taken as −40 km/h (negative because opposite direction).
4A swimmer can swim at 5 m/s in still water. A river 100 m wide flows at 3 m/s. What is the minimum time to cross the river?Crossing the River
25 s
20 s
40 s
16.7 s
Minimum time to cross a river is achieved by swimming perpendicular to the bank (regardless of drift). Time = width / vm = 100 / 5 = 20 s. The drift is (vr/vm)·w = (3/5)·100 = 60 m downstream, but the crossing time is minimised. Option A (25 s) corresponds to the shortest-distance crossing strategy where t = w/√(vm²−vr²) = 100/√(25−9) = 100/4 = 25 s. Option C incorrectly uses w/vr. Option D incorrectly uses w/(vm+vr).
5A satellite orbits from east-to-west at 8 km/s. The Earth's surface rotates at 0.5 km/s (west-to-east). What is the velocity of the satellite relative to a person on the Earth's surface?Relative velocity of Satellite
7.5 km/s
8.5 km/s
8 km/s
16 km/s
The satellite moves east-to-west (opposite to Earth's rotation). Using relative velocity formula: vse = vs + ve (speeds add for opposite directions) = 8 + 0.5 = 8.5 km/s. Option A (7.5 km/s) would be correct if the satellite moved in the same direction as Earth's rotation (wie = vs − ve = 8 − 0.5 = 7.5 km/s). Option C ignores Earth's rotation. Option D doubles the satellite speed — an arithmetic error. East-to-west orbits always have higher relative velocity to surface observers.
Practice Problems — Relative Velocity
Click "Reveal Answer" after attempting1Car A moves east at 72 km/h. Car B moves west at 54 km/h. At t=0 they are 500 m apart. After how many seconds do they meet?
10 s
12 s
14 s
16 s
👁 Reveal Answer
Correct: 10 s. Relative velocity when moving in opposite directions = 72 + 54 = 126 km/h = 35 m/s. Time = distance/relative speed = 500/35 ≈ 14.3 s. Wait — re-checking: 500/35 = 14.3 s → that gives Option C = 14 s for this specific problem. Step-by-step: v_relative = 72 + 54 = 126 km/h = 126 × (1000/3600) = 35 m/s. t = 500 m / 35 m·s⁻¹ ≈ 14.3 s ≈ 14 s.
2Rain falls at 20 m/s vertically and a cyclist moves at 20√3 m/s horizontally. Find the speed of rain as seen by the cyclist and the angle with the vertical.
40 m/s, 30°
40 m/s, 60°
20√(1+3) m/s, 30°
40 m/s, 45°
👁 Reveal Answer
Correct: 40 m/s, 60°. Speed of rain relative to cyclist: vRM = √(vR² + vM²) = √(400 + 1200) = √1600 = 40 m/s. Angle with vertical: tanθ = vM/vR = 20√3/20 = √3, so θ = 60°. The cyclist must tilt umbrella at 60° forward. Option D (45°) applies when vR = vM; here vM > vR so the angle is larger than 45°.
3A river 200 m wide flows at 4 m/s. A swimmer's speed in still water is 5 m/s. To cross in minimum distance, at what angle (with upstream bank direction) should the swimmer swim?
cos⁻¹(4/5) = 36.9° upstream
sin⁻¹(4/5) upstream
90° (perpendicular)
tan⁻¹(4/5) downstream
👁 Reveal Answer
Correct: cos⁻¹(4/5) ≈ 36.87° upstream. For minimum distance crossing, the swimmer aims upstream at angle θ where cosθ = vr/vm = 4/5. So θ = cos⁻¹(0.8) ≈ 36.9° with the upstream direction. The resulting velocity is perpendicular to banks with magnitude √(vm² − vr²) = √(25−16) = 3 m/s. Minimum distance = exactly 200 m. Time = 200/3 ≈ 66.7 s. Note: this is NOT 90° (which gives minimum time, not minimum distance).
4Two cyclists A and B move at 36 km/h and 54 km/h in the same direction. At t=0, A is 200 m ahead of B. How long does it take B to overtake A?
30 s
36 s
40 s
72 s
👁 Reveal Answer
Correct: 72 s. B is faster, so relative velocity of B w.r.t. A = 54 − 36 = 18 km/h = 5 m/s. B needs to close 200 m at 5 m/s: t = 200/5 = 40 s. So correct answer is 40 s (Option C). Verification: In 40 s, B covers 54×(40/3600)×1000 = 600 m total. A covers 36×(40/3600)×1000 = 400 m from B's start, but started 200 m ahead, so total = 600 m. They meet — confirmed.
Physics — Motion In One Dimension Revision Checklist
Check off chapters as you revise
Use this section for quick chapter tracking before mocks, part tests, and final NEET revision.
FAQ — Relative Velocity
Notes · Downloads · Revision · Important QuestionsWhat is the difference between v₁₂ and v₂₁?
v₁₂ = v₁ − v₂ (velocity of object 1 relative to object 2, pointing from 2 toward 1's perspective). v₂₁ = v₂ − v₁ = −v₁₂. They are equal in magnitude but opposite in direction. NEET questions specify which is asked — always subtract in the stated order.
Why do we add velocities when two objects move in opposite directions?
Because v₁₂ = v₁ − v₂. If v₁ is positive (say, +60 km/h east) and v₂ is negative (−40 km/h west = −40 km/h), then v₁₂ = 60 − (−40) = 100 km/h. The algebraic subtraction of a negative quantity produces a larger magnitude — mathematically equivalent to adding the magnitudes when directions are opposite.
For minimum distance river crossing, why is the condition cosθ = vr/vm and not sinθ = vr/vm?
The angle θ is measured from the upstream direction (i.e., angle between swimmer's aimed direction and the upstream bank direction). For the resultant velocity to be perpendicular to the banks, the horizontal (along-bank) component of the swimmer's velocity must cancel the river current. The component along the bank = vm cosθ = vr, giving cosθ = vr/vm. If θ were measured from the bank (not upstream), then sinθ = vr/vm — the formula changes depending on which angle convention is used.
What is the minimum time to cross a river, and does it depend on river current speed?
Minimum time t_min = w/vm where w is the river width and vm is the swimmer's speed in still water. This does NOT depend on the river current speed vr because the minimum time strategy requires swimming perpendicular to the bank: the full vm component goes toward crossing. The river current causes drift (= vr × t_min = vr·w/vm) but does not affect the crossing time.
At what NEET question format does relative velocity appear most?
Most commonly: (1) a numerical giving two velocities (direction + magnitude) and asking relative velocity of one with respect to the other; (2) rain-umbrella angle given speed of rain and observer, asking for the angle with vertical; (3) river-crossing problem asking for time or drift. Questions are typically single-calculation, testing whether you use | v1 − v2 | or v1 + v2 correctly.
If a boat rows perpendicular to the bank in a river, does it reach the opposite bank at a point directly opposite its starting point?
No. Rowing perpendicular minimises time but not distance. The boat is carried downstream by the river current by a distance = vr × (w/vm) meters. To reach the directly opposite point (minimum distance path), the boat must aim upstream at angle θ = cos⁻¹(vr/vm) so the river current cancels the along-bank component of the boat's velocity.
What is the velocity of rain relative to a stationary observer?
If the observer is stationary (vM = 0), then the velocity of rain relative to the observer equals the actual velocity of rain relative to the ground: vRM = vR − 0 = vR. The rain appears to fall exactly vertically. Only when the observer moves horizontally does the rain appear to fall at an angle from the vertical.
Can relative velocity be zero for two moving objects?
Yes. If two objects move with the same velocity vector (same speed and same direction), then v₁₂ = v₁ − v₂ = 0. An example is two cars moving at 60 km/h in the same direction — each appears stationary to the other. Geostationary satellites achieve zero velocity relative to Earth's surface by matching Earth's angular velocity.
For NRI / OCI / U.S.-Based Families
NEET NRI Counseling & Admission eBook Download
A practical guide covering sponsor rules, document checklist, verification traps, NRI quota reality, and step-by-step counselling flow. Designed to prevent last-minute rejections and wrong choice filling.
Sponsor + Proof ClarityDocuments ChecklistState-wise Traps
Tip: Keep this eBook open during verification + choice filling week for quick cross-checking.
NEET Prep (India + NRI-USA)
Schedule Trial Session For NEET Prep
Get a short diagnostic + study roadmap: syllabus gaps (NCERT vs U.S. curriculum), weak chapters, and the exact weekly plan needed to improve accuracy under time.
Gap MappingWeekly PlanAccuracy Fix
Best for: Students in Grade 10–12 (U.S. / India) who want a clear NEET timeline and daily practice structure.