Relations Uniform Accel

8 MCQs9-step worked example
Source: NCERT KinematicsPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

Lesson

The three kinematic equations for uniformly accelerated motion — v = v₀ + at, x − x₀ = v₀t + ½at², and v² = v₀² + 2a(x − x₀) — are among the most frequently tested tools in NEET physics. They look simple. The traps are not in the formulas themselves but in when and how you apply them.

The non-negotiable prerequisite: constant acceleration. Every one of these equations assumes acceleration is constant in both magnitude and direction (NCERT Class 11 Physics Chapter 2, pages 4–6). If a problem states that acceleration varies with time or position — for example, a = 2t or a = kx — these equations are invalid. You must integrate. A common NEET mistake is plugging a non-constant acceleration into v = v₀ + at and getting a plausible-looking but wrong answer.

Trap: dropping the initial velocity. When a problem says an object is "thrown downward" or "projected with initial speed u," that u is non-zero and must appear in your equation. Writing v² = 2gh instead of v² = u² + 2gh costs you the full 4 marks plus the −1 penalty. The word "thrown" is your signal: u ≠ 0.

Trap: linear thinking under uniform deceleration. A bullet entering a block slows from u to u/3 over distance d. How much farther to stop? Students instinctively scale distances linearly with speed. But v² is the quantity linear in distance, not v. Use v² = u² − 2as for each stage with the same deceleration a.

Trap: implicit-function kinematics. When time is given as a function of position (t = f(x) instead of x = f(t)), do not try to invert. Differentiate directly: v = dx/dt = 1/(dt/dx), then a = v·dv/dx. The chain rule is your friend here.

Galileo's odd-number rule. Distances in successive equal time intervals from rest follow 1 : 3 : 5 : 7, not 1 : 2 : 3 : 4. This follows directly from x = ½gt²: the nth-interval distance is proportional to (2n − 1).


Practice MCQs

Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.

MCQ 1Direct ApplicationPractice

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:

MCQ 2Easy RecallPractice

An object starts from rest and undergoes uniform acceleration. The ratio of distances covered in the 1st, 2nd, and 3rd seconds of motion is:

MCQ 3CalculationPractice

The time t (in seconds) at which a particle is at position x (in metres) is given by t = x² + x. The acceleration of the particle at x = 1 m is:

MCQ 4CalculationPractice

A bullet travelling at 360 m/s penetrates a wooden block and its speed reduces to 120 m/s after passing through 20 cm of wood. Assuming uniform deceleration, the additional thickness of wood needed to stop the bullet completely is:

MCQ 5Direct ApplicationPractice

A car accelerates uniformly from rest to 20 m/s in 10 s. The distance covered by the car during this time is:

MCQ 6Direct ApplicationPractice

A particle has velocity v₀ at t = 0 and decelerates at a constant rate of magnitude a. The distance it travels before coming to rest is:

MCQ 7Direct ApplicationPractice

A body starts from rest and moves with uniform acceleration. If it covers 5 m in the first second, the distance it covers in the 4th second is:

MCQ 8Easy RecallPractice

The kinematic equations v = v₀ + at and s = v₀t + ½at² are valid when:

Worked Example

Pattern: Multi-stage deceleration (bullet-block) — NEET pattern: multi stage deceleration bullet block

  1. 1

    Given

    - Initial speed: u = 400 m/s - Speed after first stage: v₁ = 200 m/s - Distance of first stage: d₁ = 10 cm = 0.10 m - Deceleration is uniform (constant a throughout)

  2. 2

    Required

    Additional penetration distance d₂ from v₁ = 200 m/s to rest (v₂ = 0).

  3. 3

    Concept

    Under constant deceleration, v² is linear in displacement. The third kinematic equation connects speeds to distances without needing time.

  4. 4

    Formula

    v² = u² − 2a·d (taking deceleration as positive magnitude with minus sign explicit)

  5. 5

    Substitution

    **First stage:** 200² = 400² − 2a(0.10) **Second stage:** 0² = 200² − 2a·d₂

  6. 6

    Calculation

    First stage: 40000 = 160000 − 0.20a → 0.20a = 120000 → a = 600000 m/s² Second stage: 0 = 40000 − 2(600000)d₂ → d₂ = 40000/1200000 = 1/30 m ≈ 0.0333 m = 3.33 cm **Note on exact constants:** The numbers 400, 200, and 10 cm are given problem values and are treated as exact. The factor 2 in the kinematic equation is a mathematical constant. Neither constrains significant figures in this calculation.

  7. 7

    Final answer

    d₂ = 10/3 cm ≈ 3.33 cm The bullet penetrates an additional 3.33 cm before stopping. The total penetration is 10 + 3.33 = 13.33 cm.

  8. 8

    Common trap

    The temptation is to reason: "Speed halved from 400 to 200 in 10 cm, so it should halve again in another 10 cm and reach 100, then another 10 cm to reach 50..." — this linear-speed reasoning is wrong because v² (not v) decreases linearly with distance. The remaining KE at 200 m/s is only (200/400)² = 1/4 of the original, so only 1/4 of the remaining-distance budget is left after the initial 10 cm would be wrong too — the correct remaining fraction is v₁²/(u²) of the original KE consumed, leaving d₂ = (v₁²/u²) × (u²/(2a)) ... the safest approach is to compute a from stage 1 and use it directly.

  9. 9

    Similar NEET-style question

    A car braking uniformly from 60 m/s has its speed reduced to 20 m/s after covering 80 m. What additional distance will it cover before coming to rest? *(Answer: Using the same method, a = (3600 − 400)/(2 × 80) = 20 m/s². Then d₂ = 400/(2 × 20) = 10 m.)* ---

Before solving, remember these

x − x₀ = v₀ t + ½ a t². Displacement equals initial-velocity-times-time plus half acceleration-times-time-squared, for uniform acceleration.

-- NCERT Class 11 Physics, Ch. 2, p. 5

v² = v₀² + 2 a (x − x₀). Velocity-squared relation, useful when time is not known.

-- NCERT Class 11 Physics, Ch. 2, p. 6

When brakes apply a constant deceleration -a to a vehicle with initial velocity v₀, the stopping distance is d = v₀² / (2a). Derived from v² = v₀² + 2(-a) d with v = 0.

-- NCERT Class 11 Physics, Ch. 2, p. 8

Formulas

6 formulas — click to collapse

First kinematic equation (uniform acceleration)

Final velocity equals initial velocity plus acceleration times the time elapsed, for motion under constant acceleration.

SymbolQuantitySI Unit
vFinal velocitym/s
v0Initial velocitym/s
aConstant (uniform) accelerationm/s^2
tElapsed times

Valid when

  • Acceleration a is CONSTANT (uniform) in both magnitude and direction
  • All quantities measured in the same inertial reference frame
  • Motion is along a straight line; signs encode direction along chosen axis

Do NOT use when

  • Acceleration changes in magnitude or direction (use a(t) integration)
  • Motion is uniformly circular at constant speed (a is centripetal, not tangential)

Second kinematic equation (displacement under uniform acceleration)

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.

SymbolQuantitySI Unit
xFinal positionm
x0Initial positionm
v0Initial velocitym/s
aConstant accelerationm/s^2
tTime elapseds

Valid when

  • Acceleration constant (magnitude and direction)
  • Sign convention consistent across x, v, a (one chosen positive direction)

Third kinematic equation (velocity-squared)

Relates final velocity to initial velocity, displacement, and acceleration without using time. Most useful when t is unknown or unwanted.

SymbolQuantitySI Unit
vFinal velocitym/s
v0Initial velocitym/s
aConstant accelerationm/s^2
x - x0Displacementm

Valid when

  • Constant acceleration
  • Use signed values for v, v0, a, and (x - x0) consistently

Do NOT use when

  • Time-dependent acceleration
  • Curvilinear motion where acceleration is not parallel to displacement

Projectile maximum height

Maximum height attained by a projectile launched at speed v0 and angle theta0 above the horizontal, measured above the launch level.

SymbolQuantitySI Unit
HMaximum height (above launch)m
v0Launch speedm/s
theta0Launch anglerad/deg
gGravitational accelerationm/s^2

Valid when

  • Air resistance neglected
  • Constant g over trajectory

Projectile horizontal range

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.

SymbolQuantitySI Unit
RHorizontal rangem
v0Launch speedm/s
theta0Launch angle above horizontalrad (or deg with sin in deg)
gGravitational accelerationm/s^2

Valid when

  • Launch and landing are at the same vertical height
  • Air resistance neglected
  • g treated as constant over the trajectory

Do NOT use when

  • Launch and landing heights differ (use full kinematics)
  • Significant air drag (e.g. table-tennis ball, badminton shuttle)
  • Variation of g (ballistic trajectories spanning large altitude changes)

Centripetal acceleration in uniform circular motion

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.

SymbolQuantitySI Unit
a_cCentripetal accelerationm/s^2
vTangential speedm/s
rRadius of circlem
omegaAngular speedrad/s

Valid when

  • Speed v is constant (uniform circular motion)
  • r and the centre are well-defined (instantaneous radius of curvature for general curved motion)

Do NOT use when

  • Non-uniform circular motion (then there is also a tangential acceleration component)

Exam Traps & Common Mistakes

These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.

14 items — click to collapse

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).

When it triggers

Question asks about ratios of distances traversed in successive 1-s intervals during free fall from rest.

How to avoid

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.

When it triggers

Question gives an angle the x-t line makes with the t-axis (often 30°, 45°, 60°) and asks for velocity or its ratio.

How to avoid

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.

When it triggers

Question gives t as function of x (instead of x as function of t), e.g. t = x² + x.

How to avoid

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.

When it triggers

Question phrases: 'thrown vertically downward', 'projected with initial velocity', 'launched with speed u'. If u is given numerically, it MUST appear in the equation.

How to avoid

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.

When it triggers

Question mentions launch speed and angle and asks for max height (H) or range (R). Distractors include the wrong formula's answer.

How to avoid

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).

When it triggers

Question phrases like 'thrown at angle θ with the vertical direction' or 'with horizontal'.

How to avoid

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.

When it triggers

Question describes two objects moving on the same line; observer somewhere between or alongside.

How to avoid

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.

When it triggers

Question describes a body decelerating through stages with given speed at one stage; asks for distance to stop or speed at another stage.

How to avoid

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.

When it triggers

Question describes a particle in UCM with given (R, T) then says 'now launched vertically up with same speed; find max height'.

How to avoid

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).

When it triggers

Question asks 'in uniform circular motion at constant speed, which is also constant?'

How to avoid

In UCM: SPEED constant; KE constant. VELOCITY (vector) NOT constant. ACCELERATION (centripetal, magnitude v²/r) constant in MAGNITUDE but NOT in direction.

Root cause: formula misuse

Correction

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.

Wrong option pattern

Distractor uses constant-acceleration kinematic equations on a problem where the question explicitly says acceleration changes with time or position.

Root cause: concept gap

Correction

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.

Wrong option pattern

Distractor applies R = v0^2 sin(2*theta)/g to a projectile launched from a cliff.

Past Year Questions

10 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse
NEET 2025

Two cities X and Y are connected by a regular bus service with a bus leaving in either direction every T min. A girl is driving scooty with a speed of 60 km/h in the direction X to Y notices that a bus goes past her every 30 minutes in the direction of her motion, and every 10 minutes in the opposite direction. Choose the correct option for the period T of the bus service and the speed (assumed constant) of the buses.

115 min, 120 km/h
29 min, 40 km/h
325 min, 100 km/h
410 min, 90 km/h
NTA Answer: Option 1(final)
NEET 2021

A particle moving in a circle of radius R with a uniform speed takes a time T to complete one revolution. If this particle were projected with the 'θ' same speed at an angle to the horizontal, the maximum height attained by it equals 4R. The θ, angle of projection, is then given by : 1 2gT 2 2 θ=sin−1 

1π2  R  1  2  2 gT θ=cos−1
2  π2 R 1 π2  2 R θ=cos−1 
32 gT  1 π2  2 R θ=sin−1 
42 gT 
NTA Answer: Option 1(final)

How NEET usually asks this

10 recurring patterns from past papers — click to collapse

A projectile (typically a bullet) penetrates a uniform medium with constant decelerating force; given initial speed and speed after a known distance, find the total stopping distance. Apply v² = u² + 2as to each segment, noting that 'a' is the same throughout. Common shape: bullet hits block at u, slows to u/k after distance d₁; how much further to stop?

Multi StepMedium

Common distractors

treats speed ratio as distance ratio

Linear thinking: 'speed went from u to u/3, so distance traversed should be 3× the original'

Object given a non-zero downward initial velocity from elevation; asked for the height fallen, time of impact, or final speed. Apply v² = u² + 2gh (or analogous) with proper signs. Common shape: a ball thrown vertically downward from a tower with initial speed u, hitting the ground at speed v; find the tower height. Distractors test (i) sign-of-u confusion, (ii) using v² = 2gh forgetting u², (iii) wrong g unit.

Multi StepEasy

Common distractors

drops initial velocity term

Student conflates 'thrown' with 'dropped' and uses v² = 2gh

Test yourself on this topic with real past-paper questions:

Practice this topic →

Free NEET study resources

Get a structured 30-day Mechanics plan and a complete formula booklet — delivered to your inbox instantly.