Vertical Motion and Free Fall: How Gravity Shapes Falling Objects
JEE/NEET Physics · Motion in a Straight Line series · Part 4 of 6 · All parts →
- In free fall (air resistance ignored), everything accelerates at g ≈ 9.8 m/s² downward — mass doesn’t matter
- The three kinematic equations survive intact with a = −g (up-positive convention) or a = +g (down-positive convention)
- Up-and-down trips are perfectly symmetric: rise time = fall time, and launch speed = landing speed
- Maximum height: v = 0 there, so H = u²/2g
- Signs are everything: pick up as positive once, then g enters every equation as −9.8, and any landing below the start point carries a negative displacement
Drop a coin and a key together: they hit the ground together. Heavy or light, everything falls with the same acceleration — gravity’s great democracy. This is Part 4 of the Motion in a Straight Line series, and it takes the three golden equations from Part 2 into the one situation where they do their most elegant work: vertical motion under gravity alone.
- Free fall: the mass surprise
- Throwing up: the symmetric trip
- Maximum height and time of flight
- The sign convention that saves you
- Solved examples
- Common mistakes
- This physics in your daily life
- Practice set
- Recap
Free Fall: The Mass Surprise
Here’s the puzzle that tripped up scientists for nearly two thousand years. Surely a heavier object, feeling a stronger gravitational pull, should fall faster? Gravity does pull harder on heavier things — the force is mg, so doubling the mass doubles the pull. But heavier things are also harder to accelerate: the same Newton’s second law (ma) says that doubling the mass doubles the resistance to acceleration. The two effects cancel exactly:
a = F/m = mg/m = g, for every object, every mass, every time.
A feather loses a race against a coin only because air drag cheats — it has nothing to do with gravity’s preference. Remove the air, and the feather keeps perfect pace with a hammer. Apollo 15 commander David Scott proved exactly this on the Moon in 1971, dropping a hammer and a falcon feather side by side; both hit the lunar dust simultaneously, with millions watching on live television.
What “free fall” actually means: motion under gravity alone, with air resistance ignored. It does not require the object to be falling downward from rest — a ball thrown upward at 30 m/s is in free fall for its entire flight, from the instant it leaves your hand to the instant it lands.
Throwing Up: The Symmetric Trip
Toss a ball upward at speed u. Gravity decelerates it at g every second it climbs, so it rises more and more slowly, stops for a single instant at the top (v = 0), and then falls back — gaining speed at exactly the same rate it lost it. Crucially, even at the top, when the ball is momentarily at rest, its acceleration is STILL g downward. Gravity never takes a break; only the velocity pauses.
This one-way traffic of acceleration produces a beautiful symmetry:
- Rise time = fall time — the ball takes exactly as long to climb to the top as to fall back from it.
- Landing speed = launch speed — it returns to your hand at precisely the speed you threw it (ignoring air).
- Speed at any height is the same going up and coming down — a ball passes the 10 m mark at the same speed on both legs of the trip.
The top of the flight is a pause, not a hover. The ball is not “floating” at its peak — it is simply in transition between moving up and moving down, spending zero time there.
Maximum Height and Time of Flight
All three results drop straight out of the kinematic equations by imposing one condition: at the top, v = 0.
How each one is derived, in one line each:
- Time to top: v = u + at with v = 0 gives t = u/g.
- Maximum height: v² = u² + 2as with v = 0 gives H = u²/2g.
- Total flight time (landing back at launch level): by symmetry, T = 2 × u/g = 2u/g.
Notice what these formulas reveal: double the launch speed and you quadruple the maximum height (H ∝ u²), but only double the flight time. That square is worth remembering — it appears again and again in exam questions.
| Letter | What it means (plain words) | Value / unit |
|---|---|---|
| g | free-fall acceleration near Earth’s surface | ≈ 9.8 m/s², always DOWN (sign is yours to set) |
| H | maximum height above launch | m |
| u | launch (initial) speed upward | m/s |
A practical note on the value of g: it varies slightly across the Earth — about 9.78 m/s² at the equator versus 9.83 m/s² at the poles — because the Earth bulges and rotates. JEE and NEET questions almost always specify g = 10 m/s² for clean arithmetic; use 9.8 m/s² only when the question demands it.
The Sign Convention That Saves You
Vertical motion is one-dimensional, but the two directions are physically very different, so signs matter enormously. Choose UP as positive (the usual choice for tossed objects). Then:
- Acceleration a = −9.8 m/s² always, on the way up and on the way down — never flip it mid-problem.
- Upward velocities are positive on the way up, negative on the way down.
- Any point below the launch position has negative displacement s.
With this convention locked in, all three golden equations work unchanged — no special free-fall formulas needed. Alternatively, for pure downward falls from a height, choosing DOWN as positive makes a = +g and every quantity comes out positive, which many students find cleaner. Either convention works; the fatal error is switching halfway through a problem.
Most vertical-motion errors are sign errors, not physics errors. If your answer comes out negative and the question asks for a time or a speed, check the magnitude — a negative time usually signals a sign slip or an extraneous quadratic root, as the cliff example below shows.
Solved Examples
From rest means u = 0. Velocity: v = u + gt = 0 + 10 × 3 = 30 m/s. Distance: s = ut + ½gt² = 0 + ½ × 10 × 9 = 45 m.
Notice the distance grows as t² — tripling the time doesn’t triple the fall, it nines it. ✔
Answer: 30 m/s; 45 m
H = u²/2g = 400/20 = 20 m; t_up = u/g = 20/10 = 2 s; full flight = 2 × 2 = 4 s.
Landing speed = √(2gH) = 20 m/s again — perfect symmetry, exactly as promised. ✔
Answer: H = 20 m; total flight 4 s
Down positive: u = −15 m/s (it moves against our chosen axis), a = +10 m/s², and the ground is at s = +20 m relative to the launch point.
20 = −15t + 5t² → t² − 3t − 4 = 0 → (t − 4)(t + 1) = 0 → t = 4 s (the root t = −1 s is rejected).
4 s — the negative root is physics asking to be thrown backwards in time. The ball spends part of the trip above the cliff and the rest below; one equation, one sign convention, one clean answer. ✔
Answer: 4 s
- “Heavy things fall faster.” No — mass cancels out completely; only air resistance (ignored in free fall) breaks the tie. The hammer-and-feather Moon drop settles it.
- “v = 0 at the top, so a = 0.” Wrong — velocity pauses for an instant, but acceleration remains a full g downward throughout the flight. If a were zero at the top, the ball would simply hang there forever.
- Using +g while calling up positive. With up positive, g enters every equation as −9.8 m/s²; mixing sign conventions flips answers silently. Fix your convention before writing a single equation, and never change it mid-problem.
- Forgetting the displacement sign below the launch point. A ball landing under its start point has negative s (up-positive convention) — the equation only knows where the ground is if you tell it correctly.
- Applying H = u²/2g when landing below launch level. This formula assumes the object stops at the top. If the question involves a cliff or a pit, go back to the full equation v² = u² + 2as or s = ut + ½at².
This Physics in Your Daily Life
- Water from a tap beads into droplets that accelerate as they fall — watch a thin stream break up: the widening spacing between successive drops is a live ½gt² graph drawn in water.
- Catching a high ball stings more than a low one — landing speed grows with fall height as √(2gh), so a ball dropped from four times the height arrives at only twice the speed but with four times the kinetic energy: your hands are solving v² = u² + 2as.
- Amusement park drop-towers engineer near-g free fall: that stomach-lift sensation IS your body briefly in free fall, with everything inside you falling together at the same rate.
- Cricket and baseball fielders taking high catches start moving BEFORE the ball peaks — because rise time = fall time tells them exactly when it will come down and where.
- Fountain designers tune jet speed u to reach a desired height u²/2g: every fountain arc you’ve admired is this card performed in water.
- Stunt calculators and safety nets — engineers use s = ½gt² to predict fall distances from towers and bridges, which is why airbags and crash mats are sized so precisely.
Galileo’s legend: drop a cannonball and a musket ball from the Tower of Pisa — they land together. Gravity writes a bigger pull on the cannonball, but inertia writes an equally bigger refusal, in exactly the same proportion. The two effects cancel perfectly; the universe doesn’t check your weight before accelerating you.
Numbers: g = 9.8 m/s² for all. After 1 s of falling: 9.8 m/s; after 2 s: 19.6; after 3 s: 29.4 — every falling object runs this exact velocity schedule, pebble or boulder. Distance fallen: 4.9, 19.6, 44.1 m — one timetable for the whole planet.
Draw the v–t graph for a toss: a straight line tilting down through zero — upward velocity bleeding away, zero at the top (a single instant, not a shelf), then negative and growing. The SAME straight line covers both rise and fall: one slope (−g), one picture, symmetric on either side of the zero-crossing.
Practice Set (Answers Hidden — Try First)
(NEET-level) A ball is dropped from rest. Its speed after 2 s (g = 10 m/s²) is:
(JEE Main-level) A ball is thrown up at u = 30 m/s. Time to return to the thrower’s hand:
(NEET-level) Maximum height reached when u = 20 m/s (g = 10):
(Concept) At the highest point of a throw, the ball’s velocity and acceleration are:
(JEE Main-level) Time to fall freely from 45 m (g = 10 m/s²):
(JEE Advanced flavour) A ball thrown up passes a window on the way up and again on the way down. Compare its speeds at the two crossings:
- Free fall: a = g for ALL masses
- Up-positive convention: a = −9.8 throughout, no exceptions
- Top of flight: v = 0, but a = g still
- H = u²/2g, T_flight = 2u/g (same level)
- Symmetry: rise time = fall time, launch speed = landing speed
- 🔁 all masses fall at g
- 🔁 v = 0 but a = g at the top
- 🔁 H = u²/2g · T = 2u/g
Frequently Asked Questions
Q. Does a heavier object ever fall faster in reality? Yes — but only because of air resistance, not gravity. A sheet of paper flutters down slowly because drag fights its small weight; crumple it into a ball and it falls much faster with the same mass. In free-fall problems, air is ignored and mass never appears.
Q. Is the acceleration zero at the highest point? Never. v = 0 at the top for one instant, but a = g downward at every moment of the flight. Zero velocity with nonzero acceleration is exactly what a ball at its peak — or a car braking to a halt — looks like.
Q. Why do I get a negative time in some problems? A quadratic in t often yields two mathematical roots. The negative root has no physical meaning here — it’s the time the particle would have passed that position if its motion had extended backwards in time. Keep only the positive root, unless a later part of the problem says otherwise.
Q. Can I use g = 10 m/s² in exams? Yes when the question states it, which is most NEET and JEE Main questions. Use 9.8 m/s² when the question explicitly gives it or when options are finely separated.
Recap
- 🧠 Chant: ‘gravity never rests — even at the top’.
- 🧠 Symmetry: ‘the up-mirror is the down-mirror’ — rise time = fall time, launch speed = landing speed.
- 🧠 Mass never appears in a free-fall equation: a = mg/m = g for everyone.
- 🧠 g ≈ 10 for quick maths, 9.8 for accuracy — whichever the question specifies.
- 🔧 Method: fix a sign convention before writing equations, and never switch mid-problem.
- 🏠 Daily: tap-water drop spacing = the ½gt² graph, live in your kitchen.
- 🏠 Daily: fielders move early for high catches because rise time = fall time.
Next up: Part 5 asks what happens when two objects move — and motion quietly starts depending on where you stand. With the sign discipline you built here, relative velocity will feel like a natural next step.
Quick revision
- In free fall (air resistance ignored), everything accelerates at g ≈ 9.8 m/s² downward — mass doesn’t matter
- The three kinematic equations survive intact with a = −g (up-positive convention) or a = +g (down-positive convention)
- Up-and-down trips are perfectly symmetric: rise time = fall time, and launch speed = landing speed
- Maximum height: v = 0 there, so H = u²/2g
- Signs are everything: pick up as positive once, then g enters every equation as −9.8, and any landing below the start point carries a negative…
- Free fall: the mass surprise
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