Velocity Time Graph

8 MCQs3 revision cards9-step worked example
Source: NCERT KinematicsPYQ coverage: NEET 2020, 2021, 2022, 2023, 2024, 2025Official key: NTA-verifiedLast reviewed: May 2026

Lesson

The velocity-time (v-t) graph is one of the most information-dense representations in kinematics. NEET questions built on it test whether you can extract three quantities correctly: velocity at an instant (read the y-value), acceleration (compute the slope), and displacement (compute the area under the curve). The common trap is treating the graph like a position-time graph and misreading what slope and area represent.

What the v-t graph encodes. For straight-line motion under uniform acceleration, v = v₀ + at (NCERT Class 11 Physics Chapter 2, page 5). On the v-t graph this is a straight line whose slope equals the constant acceleration a, and whose y-intercept is the initial velocity v₀. A horizontal line means zero acceleration (uniform motion). A line sloping downward means deceleration (negative a if positive direction is upward).

Displacement as area. The displacement between times t₁ and t₂ equals the area under the v-t curve between those times. For constant acceleration, the shape is a trapezoid (or triangle if v₀ = 0). The second kinematic equation s = v₀t + ½at² is exactly this area. A high-frequency NEET trap: for free fall from rest, successive-second distances follow the odd-number ratio 1 : 3 : 5 : 7, not the linear ratio 1 : 2 : 3 : 4. The quadratic dependence of displacement on time (s = ½gt²) makes equal time intervals yield unequal distances.

When the graph is NOT a straight line. If the v-t curve is nonlinear, acceleration is not constant. You cannot use v = v₀ + at or v² = v₀² + 2as in that region — those three kinematic equations require constant acceleration. For a curved v-t graph, acceleration at any instant is the tangent's slope, and displacement is still the area (computed by integration or geometric estimation).

Watch-out for NEET. When a problem states an object is "thrown downward" with initial speed u, the v-t graph starts at v₀ = u (not zero). Using v² = 2gh instead of v² = u² + 2gh costs the full 4 marks plus the 1-mark negative penalty.


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 1Easy RecallPractice

The slope of a velocity-time graph for a body moving in a straight line represents:

MCQ 2Easy RecallPractice

The area under a velocity-time graph between two instants gives the:

MCQ 3Easy RecallPractice

A velocity-time graph is a horizontal straight line at v = 5 m/s. Which statement is correct?

MCQ 4Direct ApplicationPractice

A body starts from rest and accelerates uniformly. The distances covered in the 1st, 2nd, and 3rd seconds of its motion are in the ratio:

MCQ 5Direct ApplicationPractice

A ball is thrown vertically downward from a tower with an initial speed of 10 m/s. It strikes the ground with a speed of 30 m/s. Taking g = 10 m/s², the height of the tower is:

MCQ 6Direct ApplicationPractice

The v-t graph of a body is a straight line passing through the origin with a slope of 3 m/s². The displacement of the body in the first 4 seconds is:

MCQ 7Concept TrapPractice

The velocity-time graph of a particle is a curve (not a straight line) between t = 0 and t = 5 s. A student uses v = v₀ + at to find the velocity at t = 5 s. This approach is:

MCQ 8CalculationPractice

A particle starts from rest and moves with constant acceleration. If it covers 10 m in the first 2 seconds, what distance does it cover in the next 2 seconds (from t = 2 s to t = 4 s)?

Quick recall before you leave

Worked Example

  1. 1

    Given

    - Initial velocity: v₀ = 0 (released from rest) - Acceleration: a = g = 10 m/s² (constant, downward) - Required: ratio of distances in the 1st, 2nd, 3rd, and 4th individual seconds

  2. 2

    Required

    Distance covered in each successive 1-second interval: d₁, d₂, d₃, d₄ and their ratio.

  3. 3

    Concept

    Under constant acceleration from rest, cumulative distance grows as s = ½gt². The distance in the nth second equals sₙ − sₙ₋₁ = ½g(n² − (n−1)²) = ½g(2n − 1). The factor (2n − 1) produces the odd-number sequence 1, 3, 5, 7, ...

  4. 4

    Formula

    Distance in nth second: dₙ = ½g(2n − 1) This is derived from the second kinematic equation s = ½at² by subtraction of consecutive cumulative distances.

  5. 5

    Substitution

    - d₁ = ½(10)(2×1 − 1) = 5 × 1 = 5 m - d₂ = ½(10)(2×2 − 1) = 5 × 3 = 15 m - d₃ = ½(10)(2×3 − 1) = 5 × 5 = 25 m - d₄ = ½(10)(2×4 − 1) = 5 × 7 = 35 m

  6. 6

    Calculation

    Ratio d₁ : d₂ : d₃ : d₄ = 5 : 15 : 25 : 35 = 1 : 3 : 5 : 7 Note on exact constants: g = 10 m/s² is a problem-defined exact value, and the integers 1, 2, 3, 4 are counting numbers. These do not limit significant figures in the ratio.

  7. 7

    Final answer

    The distances in the 1st, 2nd, 3rd, and 4th seconds are in the ratio **1 : 3 : 5 : 7**.

  8. 8

    Common trap

    Students often answer 1 : 2 : 3 : 4, assuming distance grows linearly with time. This is wrong because distance grows as t² (quadratic), making successive-interval distances follow the odd-number rule, not a linear sequence. On the v-t graph, the area of each successive strip (between t = n−1 and t = n) is a trapezoid whose area grows as (2n − 1), not as n.

  9. 9

    Similar NEET-style question

    A stone is dropped from a tall building. If it covers 20 m in the first 2 seconds, what distance does it cover in the 3rd and 4th seconds combined? (Use the odd-number ratio for 2-second intervals, or compute directly from s = ½gt².) ---

Before solving, remember these

For motion in a straight line, the area between the velocity–time curve and the time axis, between two instants, equals the displacement of the object over that interval. This geometric interpretation is the basis for graphical derivations of the kinematic equations.

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

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

Sources

NCERT refs: Class 11 Physics Chapter 2, p.5

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