JEE/NEET Physics · Motion in a Plane series · Part 6 of 6 · All parts →
- One card compresses six parts — vectors, both projectile types, 2-D relativity, circles
- Aₓ = A cosθ, A_y = A sinθ · R_resultant = √(Σx² + Σy²)
- Horizontal launch: t = √(2h/g), R = u√(2h/g) · Angular: T, H, R toolkit
- River: straight across for time, sin⁻¹(v_s/v_b) upstream for position
- a_c = v²/r — turning is acceleration
The final card of the Motion in a Plane series — six parts on one revision sheet, plus the sports, defence and space engineering that runs on these formulas.
- The master formula card
- The one-rule-per-part recap
- 2-D motion in the wide world
- Final practice set
- Recap
The Master Formula Card
| What | Formula / rule | Remember |
|---|---|---|
| Resolution | Aₓ = A cosθ · A_y = A sinθ | cos along, sin across |
| Resultant | √(Aₓ² + A_y²) at tan⁻¹(A_y/Aₓ) | Pythagoras back up |
| Horizontal projectile | t = √(2h/g) · R = u√(2h/g) | height alone sets time |
| Angular projectile | T = 2u sinθ/g · H = u²sin²θ/2g | vertical movie with a start |
| Range | R = u² sin2θ/g | max 45°, twins θ & 90−θ |
| Velocity at top | u cosθ horizontal | never zero |
| Relative 2-D | v_A − v_B component-wise | subtract the seer |
| River crossing | t = d/v_b (⊥ aim) | opposite bank: sin⁻¹(v_s/v_b) upstream |
| Circular motion | a_c = v²/r = ω²r | ⊥ velocity, centre-pointing |
| Circle speed | v = 2πr/T | period link |
| Centripetal force | F = mv²/r | a role, not a new force |
The One-Rule-Per-Part Recap
Part 1: resolve everything onto smart axes. Part 2: horizontal and vertical motion are two independent films. Part 3: climb and push trade off — 45° balances them. Part 4: subtract the observer, component-wise. Part 5: turning velocity is acceleration, v²/r toward the centre.
2-D Motion in the Wide World
- Every artillery and missile system solves the angular-projectile equations in real time, correcting for air drag and the Earth’s rotation beneath the parabola.
- Olympic biomechanics labs tune javelin release to ~35° (drag-adjusted from 45°), shot-put ~38–41°: world records set by sin2θ management.
- ISRO’s launch windows — a rocket ascending into orbit is a projectile problem chained to circular motion: this series plus the Gravitation series built every launch.
- Banked highway curves — engineers tilt roads so that the horizontal component of the normal force supplies mv²/r, reducing the friction demanded: geometry as a safety device.
- Cricket ball-tracking (Hawk-Eye) — full 3-D vector kinematics predicting the parabolic future of the ball, adjudicating LBW within the error bars of this chapter.
One dimension was a rail track — forward or back, no choices. Two dimensions is a field: every direction, every trade-off. The moment direction became free, physics needed vectors to speak, components to compute, and geometry to decide outcomes. Everything after this — forces, fields, orbits — happens in this wider arena.
The toolkit in one breath: resolve (Part 1), let each axis run its own movie (Parts 2–3), subtract the observer when frames disagree (Part 4), and pay v²/r whenever direction turns (Part 5). A football pass uses all five in one second — your brain approximating what these formulas compute exactly.
Picture a park seen from above: a jogger’s zigzag (vectors), a thrown frisbee (projectile), a river ferry (relative motion), a cyclist circling the fountain (UCM). One aerial photo contains the whole series — six chapters, one playground.
Practice set (answers hidden — try first)
(NEET-level) 3 N east + 4 N north: resultant =
(JEE Main-level) Ball off a 45 m table at 12 m/s: range (g=10) =
(NEET-level) u = 20, θ = 45°: range =
(JEE Main-level) v = 10 on r = 2.5: a_c =
(NEET-level) Boat 6 ⊥ stream 8: resultant =
- resolution + Pythagoras = the vector cycle
- projectiles: independence of axes does everything
- 45° max range; twins share a range
- relative motion: subtract component-wise
- circles: a_c = v²/r, turning = acceleration
- 🔁 the 11-row master card
- 🔁 one rule per part
- 🔁 2-D motion = the arena of all later physics
- 🧠 Full-card chant: ‘resolve, release, trade, subtract, turn’.
- 🏠 Daily: banked curves supply mv²/r by geometry.
- 🏠 Daily: Hawk-Eye adjudicates with this chapter.
Quick revision
- One card compresses six parts — vectors, both projectile types, 2-D relativity, circles
- Aₓ = A cosθ, A_y = A sinθ · R_resultant = √(Σx² + Σy²)
- Horizontal launch: t = √(2h/g), R = u√(2h/g) · Angular: T, H, R toolkit
- River: straight across for time, sin⁻¹(v_s/v_b) upstream for position
- a_c = v²/r — turning is acceleration
- The one-rule-per-part recap
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