JEE/NEET Physics · Motion in a Plane series · Part 2 of 6 · All parts →
- A horizontally-thrown ball falls EXACTLY as fast as one simply dropped — the horizontal throw changes where, not how fast, it falls
- Two independent movies at once: horizontal = constant velocity, vertical = free fall
- Time of fall depends ONLY on height: t = √(2h/g)
- Horizontal range = u × t = u√(2h/g)
- Velocity at any instant: combine vₓ (constant) and v_y = gt by Pythagoras
Famous thought experiment: fire a bullet horizontally and drop an identical bullet at the same instant — they hit the ground together. The horizontal motion and the vertical fall don’t argue; they simply ignore each other. Part 2 of the Motion in a Plane series.
- The two-movie trick
- Time of flight from height alone
- Range of the horizontal throw
- Velocity on the way down
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
The Two-Movie Trick
A ball rolled off a table is starring in two films simultaneously. Horizontal film: no horizontal force → constant velocity u, forever. Vertical film: gravity → free fall from rest, ½gt² downward. The films never interact — the total motion is just both playing at once. This independence is the entire secret of projectiles.
Time of Flight from Height Alone
Vertical film: h = ½gt² → t = √(2h/g). Notice what’s missing: the throw speed. A bullet fired at 1000 m/s and one gently nudged off the same table land at the same MOMENT — the faster one just lands much farther away.
Range of the Horizontal Throw
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| u | horizontal launch speed (constant all flight) | m/s |
| h | height of the launch point above the floor | m |
| t | time in the air (set by h alone) | s |
| R | horizontal distance travelled before landing | m |
| v_y | downward speed so far: gt | m/s |
Velocity on the Way Down
At any instant: vₓ = u (never changes) and v_y = gt (grows steadily). The actual velocity is their vector sum — speed √(u² + g²t²), dipping steeper and steeper. The path traced is a parabola: flat at launch, plunging at the end.
Solved Examples
t = √(2×1.25/10) = 0.5 s.
R = 4 × 0.5 = 2 m.
✔
Answer: 0.5 s; 2 m
t = √(2×20/10) = 2 s; v_y = 20 m/s at landing.
Speed = √(15² + 20²) = 25 m/s, at tan⁻¹(20/15) ≈ 53° below horizontal.
✔
Answer: 25 m/s at 53° below horizontal
t = √(2×45/10) = 3 s; package travels 40 × 3 = 120 m forward while falling.
Release 120 m before the target — this is literally how cargo drops and aid deliveries are computed.
✔
Answer: 120 m early
- Thinking a faster horizontal throw stays up longer. Flight time comes from height alone — horizontal speed has zero vote.
- Using the 1-D equation v² = u² + 2as across both axes. Each equation belongs to ONE axis; never mix x quantities into y equations.
- Forgetting vₓ never changes. No horizontal force (air ignored) means the horizontal velocity at landing equals the launch value.
- Treating the path as a straight slanted line. It’s a parabola — flat start, accelerating dip; the steepness grows with time, not distance.
This Physics in Your Daily Life
- Basketball passes are aimed above and ahead — your brain runs the two-movie trick: lead the runner (horizontal constancy) while gravity pulls the ball (vertical fall).
- Cargo/aid drops from planes release hundreds of metres before the target: R = u√(2h/g), exactly Part 2’s drone example with lives at stake.
- Water from a horizontally-held hose — the jet’s parabola is this card drawn in water; bend the hose upward and you’re halfway to Part 3.
- Long-jump take-off boards — jumpers fight for height because time aloft (√(2h/g)) is what converts speed into distance.
- Monkey-and-hunter demonstrations — a dart aimed AT a dropped monkey hits it: gravity affects both identically, the classroom classic of independence of motion.
Two channels on one screen: the horizontal channel plays ‘constant speed cruise’, the vertical channel plays ‘free fall’. Neither director watches the other’s film. The ball doesn’t ‘know’ it’s moving sideways as it falls, and doesn’t ‘know’ it’s falling as it cruises. Independence, not cooperation.
Ball off a 45 m cliff at 10 m/s: every 0.5 s it has moved exactly 5 m across (10 × 0.5) while falling 1.25, 5, 11.25 m (½gt²) — the across-column counts evenly, the down-column squares. Read the two columns side by side and you’re reading the parabola’s DNA.
Plot the trail: mark equal horizontal steps (cruise), and at each mark drop by ½gt² (fall). Equal steps with squaring drops trace a parabola — flat at launch, steeper later. One graph, both movies superimposed.
Practice set (answers hidden — try first)
(NEET-level) Ball off a 45 m table: flight time (g=10) =
(JEE Main-level) Off a 20 m cliff at 10 m/s: range =
(NEET-level) Doubling horizontal speed from a fixed height:
(Concept) A bullet fired horizontally and one dropped simultaneously from the same height:
(JEE Main-level) vₓ = 12, v_y at landing = 16: landing speed =
- two independent movies: cruise + fall
- t = √(2h/g) — throw speed has no vote
- R = u√(2h/g)
- vₓ constant forever; v_y = gt
- path is a parabola
- 🔁 independence of x and y motion
- 🔁 t = √(2h/g) from height alone
- 🔁 R = u√(2h/g)
- 🧠 Chant: ‘height sets time, speed sets distance’.
- 🧠 Classic: fired bullet and dropped bullet land together.
- 🏠 Daily: basketball leads the runner, gravity drops the ball.
- 🏠 Daily: aid planes release early by R = u√(2h/g).
Quick revision
- A horizontally-thrown ball falls EXACTLY as fast as one simply dropped — the horizontal throw changes where, not how fast, it falls
- Two independent movies at once: horizontal = constant velocity, vertical = free fall
- Time of fall depends ONLY on height: t = √(2h/g)
- Horizontal range = u × t = u√(2h/g)
- Velocity at any instant: combine vₓ (constant) and v_y = gt by Pythagoras
- Time of flight from height alone
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