Acceleration
Acceleration is the rate of change of velocity with time: a = dv/dt. Average acceleration over an interval is Δv/Δt. Uniform acceleration means a is constant in magnitude and direction.
-- NCERT Class 11 Physics, Ch. 2, p. 3The trap first: When a problem says an object is "thrown vertically downward with speed u," many aspirants reflexively set u = 0 and use v² = 2gh. That single dropped term costs 4 marks — the correct relation is v² = u² + 2gh. This is one of the most frequently punished errors in NEET kinematics for this topic.
What uniformly accelerated motion means. A body has uniformly accelerated motion when its acceleration remains constant in both magnitude and direction throughout the motion (NCERT Class 11 Physics Chapter 2, page 3). "Constant" means the velocity changes by equal amounts in equal time intervals. The three kinematic equations that follow from this single condition are:
These hold ONLY when acceleration is constant. If a problem states that acceleration varies with time or position, these equations cannot be applied — you must integrate (NCERT Class 11 Physics Chapter 2, page 7).
Free fall as the model case. An object in free fall near Earth's surface (air resistance negligible) has constant acceleration g ≈ 9.8 m/s² downward. From rest, the distances covered in the 1st, 2nd, 3rd, 4th seconds are in the ratio 1 : 3 : 5 : 7 — Galileo's odd-number rule — because displacement grows as t², so successive-interval distances go as (2n − 1).
The sign-convention trap. Every variable in the kinematic equations carries a sign determined by the chosen positive direction. "Thrown downward" means v₀ is nonzero and has the same sign as g. "Thrown upward" means v₀ opposes g. Forgetting to assign the correct sign to v₀ — or worse, setting it to zero — is the root cause of wrong answers in vertical-kinematics problems.
Watch-out for non-uniform scenarios. If the problem says "acceleration changes with time" or gives a = f(t), stop — the kinematic equations do not apply. Use calculus: a = dv/dt, integrate.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
An object starts from rest and falls freely under gravity. The ratio of distances covered in the 1st, 2nd, and 3rd seconds of its fall is:
A ball is thrown vertically downward from the top of a tower with an initial speed of 10 m/s. It reaches the ground with a speed of 30 m/s. Taking g = 10 m/s², the height of the tower is:
Which of the following is the correct definition of uniformly accelerated motion?
A body starts from rest and moves with uniform acceleration. The ratio of the total distance covered at the end of 1 s, 2 s, and 3 s is:
The three kinematic equations (v = v₀ + at, s = v₀t + ½at², v² = v₀² + 2as) are valid when:
A particle is given a relation t = αx² + βx, where t is time and x is position (α and β are positive constants). The acceleration of the particle is:
A stone is dropped from rest. The distance covered by it in the 4th second of its fall is (take g = 10 m/s²):
A ball is thrown vertically upward with an initial velocity of 20 m/s. Taking g = 10 m/s², the maximum height reached is:
Pattern: Vertical kinematics with nonzero initial velocity (NEET pattern: vertical kinematics initial velocity nonzero — appeared in NEET 2020, 2023).
Given
- Initial speed u = 15 m/s (downward) - Final speed v = 25 m/s (downward) - g = 10 m/s² (downward, same direction as motion)
Required
Height of building h.
Concept
Since the ball moves under constant gravitational acceleration (free fall with initial velocity), the three kinematic equations apply. We need to relate speeds and displacement without time, so the third kinematic equation is the tool.
Formula
v² = u² + 2gh (Taking downward as positive, both u and g are positive.)
Substitution
(25)² = (15)² + 2(10)h
Calculation
625 = 225 + 20h 20h = 400 h = 20 m **Note on exact constants:** g = 10 m/s² is stated as an exact problem-defined value and does not limit significant figures. The speeds 15 and 25 are treated as exact given values.
Final answer
The height of the building is 20 m.
Common trap
The most frequent error is setting u = 0 (treating "thrown downward" as "dropped"), which gives h = v²/(2g) = 625/20 = 31.25 m — a wrong answer that typically appears as a distractor. Always check: if the problem says "thrown" or "projected" with a stated speed, that speed is nonzero and must appear in the equation.
Similar NEET-style question
A stone is projected vertically downward from a cliff with a speed of 5.0 m/s and reaches the bottom with a speed of 15 m/s. If g = 10 m/s², find the height of the cliff. (Answer: v² = u² + 2gh → 225 = 25 + 20h → h = 10 m.) ---
Acceleration is the rate of change of velocity with time: a = dv/dt. Average acceleration over an interval is Δv/Δt. Uniform acceleration means a is constant in magnitude and direction.
-- NCERT Class 11 Physics, Ch. 2, p. 3Figure 2.7 shows the variation of (a) acceleration, (b) velocity, and (c) distance with time for an object in free fall. With y-axis pointing upward and a = -g, velocity decreases linearly while distance follows the quadratic y = -½ g t².
-- NCERT Class 11 Physics, Ch. 2, p. 8For an object dropped from rest, the distances traversed in successive equal intervals τ are in the ratio 1 : 3 : 5 : 7 : 9 : 11 ... (odd numbers). This ratio follows directly from y = ½ g t² applied to successive intervals.
-- NCERT Class 11 Physics, Ch. 2, p. 8Free fall near the Earth's surface (neglecting air resistance) is uniformly accelerated motion with a = -g ≈ -9.8 m/s² (downward). The kinematic equations apply with this constant a.
-- NCERT Class 11 Physics, Ch. 2, p. 7Final velocity equals initial velocity plus acceleration times the time elapsed, for motion under constant acceleration.
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | Final velocity | m/s |
| v0 | Initial velocity | m/s |
| a | Constant (uniform) acceleration | m/s^2 |
| t | Elapsed time | s |
Displacement equals initial-velocity-times-time plus half of acceleration-times-time-squared. The (1/2) factor is the area of the triangle on the v-t graph.
| Symbol | Quantity | SI Unit |
|---|---|---|
| x | Final position | m |
| x0 | Initial position | m |
| v0 | Initial velocity | m/s |
| a | Constant acceleration | m/s^2 |
| t | Time elapsed | s |
Relates final velocity to initial velocity, displacement, and acceleration without using time. Most useful when t is unknown or unwanted.
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | Final velocity | m/s |
| v0 | Initial velocity | m/s |
| a | Constant acceleration | m/s^2 |
| x - x0 | Displacement | m |
Maximum height attained by a projectile launched at speed v0 and angle theta0 above the horizontal, measured above the launch level.
| Symbol | Quantity | SI Unit |
|---|---|---|
| H | Maximum height (above launch) | m |
| v0 | Launch speed | m/s |
| theta0 | Launch angle | rad/deg |
| g | Gravitational acceleration | m/s^2 |
For a projectile launched from and returning to the same horizontal level with initial speed v0 at angle theta0 above the horizontal, the horizontal range R is given by this formula. R is maximised at theta0 = 45 deg.
| Symbol | Quantity | SI Unit |
|---|---|---|
| R | Horizontal range | m |
| v0 | Launch speed | m/s |
| theta0 | Launch angle above horizontal | rad (or deg with sin in deg) |
| g | Gravitational acceleration | m/s^2 |
An object moving in a circle of radius r at constant speed v has acceleration of magnitude v^2/r (or equivalently omega^2 * r) directed toward the centre. This is centripetal (radially inward), not tangential.
| Symbol | Quantity | SI Unit |
|---|---|---|
| a_c | Centripetal acceleration | m/s^2 |
| v | Tangential speed | m/s |
| r | Radius of circle | m |
| omega | Angular speed | rad/s |
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
Category: Similar Terms
Student answers 1:2:3:4 for distances in successive 1-second intervals (linear) instead of 1:3:5:7 (Galileo's odd numbers).
Question asks about ratios of distances traversed in successive 1-s intervals during free fall from rest.
Distance grows quadratically (y = ½ g t²); successive interval distances are y_n - y_{n-1} = ½ g (t_n² - t_{n-1}²) = ½ g (2n-1) seconds. The factor (2n-1) gives 1, 3, 5, 7, ...
Category: Graph Interpretation
Student uses sin or cos of the angle the line makes with the time axis, instead of tan, to extract velocity.
Question gives an angle the x-t line makes with the t-axis (often 30°, 45°, 60°) and asks for velocity or its ratio.
Velocity = dx/dt = slope of x-t line = tan(angle), where the angle is measured from the time axis. Always tan, not sin or cos.
Category: Overthinking
Student attempts to invert t(x) algebraically before differentiating, getting tangled in messy algebra; misses chain rule.
Question gives t as function of x (instead of x as function of t), e.g. t = x² + x.
Differentiate the given relation directly: dt/dx = (function of x). Then v = dx/dt = 1/(dt/dx). For acceleration use chain rule: a = dv/dt = (dv/dx)(dx/dt) = v dv/dx.
Category: Sign Convention
Student treats a 'thrown vertically downward' problem as if the object were dropped (u = 0). The result is wrong by an additive u² term in v² = u² + 2gh. When the question explicitly states a launch speed, that speed is non-zero and CANNOT be ignored.
Question phrases: 'thrown vertically downward', 'projected with initial velocity', 'launched with speed u'. If u is given numerically, it MUST appear in the equation.
Always parse the launch verbal cue and write down u with its sign before reaching for v² = 2gh. Use the full v² = u² + 2gh (or u² - 2gh for upward motion).
Category: Similar Terms
Student plugs into v₀² sin(2θ)/g (range) when asked for maximum height, or vice versa. The two share v₀ and θ but have different sin-vs-sin² and 2g-vs-g terms.
Question mentions launch speed and angle and asks for max height (H) or range (R). Distractors include the wrong formula's answer.
Memorise BOTH formulas explicitly: H = v₀² sin² θ / (2g) (note sin²); R = v₀² sin(2θ) / g (note sin of doubled angle). Check by setting θ = 45°: max range, half max height.
Category: Sign Convention
Student plugs angle θ into v cos θ when the question states 'angle with the vertical' (which makes the horizontal component v sin θ).
Question phrases like 'thrown at angle θ with the vertical direction' or 'with horizontal'.
Always identify reference axis explicitly. From horizontal: vx = v cos θ, vy = v sin θ. From vertical: vx = v sin θ, vy = v cos θ. The two are complementary (θ_h + θ_v = 90°).
Category: Sign Convention
Student fails to distinguish between same-direction and opposite-direction relative velocities, treating both as magnitudes.
Question describes two objects moving on the same line; observer somewhere between or alongside.
Relative velocity is a VECTOR. Same direction: v_rel = v_a - v_b (smaller magnitude). Opposite direction: v_rel = v_a + v_b (larger magnitude). Use sign convention consistently along chosen axis.
Category: Overthinking
Student assumes proportionality of speed to remaining distance under uniform deceleration. In fact, KE drops linearly with distance (v² is the linear quantity, not v): v² = u² - 2as. Speed-vs-distance is a sqrt-curve, not a line.
Question describes a body decelerating through stages with given speed at one stage; asks for distance to stop or speed at another stage.
Always work with v², not v, when uniform deceleration is in play. The work-energy theorem gives the same answer faster: ½ m v² = work done against constant force over distance.
Category: Overthinking
Student uses the radius R as the projectile launch height or fails to compute the UCM speed from period.
Question describes a particle in UCM with given (R, T) then says 'now launched vertically up with same speed; find max height'.
Step 1: speed v = 2πR/T (from UCM). Step 2: max projectile height H = v²/(2g) = (2πR/T)² / (2g). Don't shortcut by setting H = R.
Category: Similar Terms
Student claims velocity is constant in uniform circular motion (it's not — direction changes).
Question asks 'in uniform circular motion at constant speed, which is also constant?'
In UCM: SPEED constant; KE constant. VELOCITY (vector) NOT constant. ACCELERATION (centripetal, magnitude v²/r) constant in MAGNITUDE but NOT in direction.
Root cause: concept gap
Average velocity = total displacement / total time (vector). Average speed = total path length / total time (scalar, always >= |average velocity|). For round-trip motion, average velocity is zero; average speed is not.
Distractor offers (v1+v2)/2 instead of total-distance/total-time.
Root cause: formula misuse
The three kinematic equations require CONSTANT acceleration. For variable acceleration, use a = dv/dt and integrate, or use v dv = a dx for position-dependent acceleration. Verify constant-a before applying these formulas.
Distractor uses constant-acceleration kinematic equations on a problem where the question explicitly says acceleration changes with time or position.
Root cause: concept gap
Standard range R = v0^2 sin(2*theta)/g and H = v0^2 sin^2(theta)/(2g) assume (i) launch and landing at the same height, (ii) negligible air drag, and (iii) constant g. For asymmetric trajectories, use the full kinematic decomposition along x and y.
Distractor applies R = v0^2 sin(2*theta)/g to a projectile launched from a cliff.
Root cause: concept gap
Acceleration is the rate of change of VELOCITY (a vector), not speed. In uniform circular motion, the speed is constant but the velocity direction changes continuously, giving a centripetal acceleration of magnitude v^2/r toward the centre.
Distractor option says 'a = 0 because speed is constant'.
A particle moving with uniform speed in a circular path maintains:
The ratio of the distances travelled by a freely falling body in the 1st, 2nd, 3rd and 4th second
uses arithmetic progression 1 2 3 4
Linear-time intuition
treats t x as explicit
Default to t-as-input thinking
treats speed ratio as distance ratio
Linear thinking: 'speed went from u to u/3, so distance traversed should be 3× the original'
uses sin or cos instead of tan
Trig confusion under time pressure
uses radius as height
Surface confusion of geometric R and projectile H
uses sin instead of sin squared
Confusing the height formula H = v²sin²θ/(2g) with the range 2v sin θ / g
uses sin instead of cos
Sign-of-angle-axis confusion
speed zero at top
True only for vertical motion (no horizontal v₀)
ignores direction relative to observer
Treating bus-passing time as absolute
treats velocity as scalar
Conflates speed (scalar) with velocity (vector)
drops initial velocity term
Student conflates 'thrown' with 'dropped' and uses v² = 2gh
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