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. 5Velocity Time Graph
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.
The slope of a velocity-time graph for a body moving in a straight line represents:
The area under a velocity-time graph between two instants gives the:
A velocity-time graph is a horizontal straight line at v = 5 m/s. Which statement is correct?
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:
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:
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:
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:
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
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
Required
Distance covered in each successive 1-second interval: d₁, d₂, d₃, d₄ and their ratio.
- 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
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
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
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
Final answer
The distances in the 1st, 2nd, 3rd, and 4th seconds are in the ratio **1 : 3 : 5 : 7**.
- 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
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
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.
| 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 |
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.
| 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 |
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.
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | Final velocity | m/s |
| v0 | Initial velocity | m/s |
| a | Constant acceleration | m/s^2 |
| x - x0 | Displacement | m |
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.
| 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 |
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.
| 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 |
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.
| 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 |
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: concept gap
Correction
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.
Wrong option pattern
Distractor offers (v1+v2)/2 instead of total-distance/total-time.
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.
Root cause: concept gap
Correction
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.
Wrong option pattern
Distractor option says 'a = 0 because speed is constant'.
Past Year Questions
10 questions from NEET 2020, 2021, 2022, 2023, 2024, 2025. Answers verified against NTA official keys. — click to collapse
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
How NEET usually asks this
10 recurring patterns from past papers — click to collapse
For an object dropped from rest, distances traversed in successive equal time intervals are in the ratio 1:3:5:7:9:... (odd numbers). Derivable from y = ½ g t². Common shape: 'find ratio of distance covered in 1st, 2nd, 3rd, 4th seconds of free fall'.
Common distractors
uses arithmetic progression 1 2 3 4
Linear-time intuition
Time given as implicit function of position, e.g. t = x² + x. To find acceleration, differentiate twice: dt/dx = 2x + 1, so v = dx/dt = 1/(2x+1); a = dv/dt = -2v²·v = -2v³ (chain rule). Multi-step calculus.
Common distractors
treats t x as explicit
Default to t-as-input thinking
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?
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'
Two particles' position-time graphs are straight lines with given angles vs the time axis (e.g. 30° and 45°); find ratio of velocities. Velocity = slope = tan(angle). Common shape: ratio = tan(30°)/tan(45°) = 1/√3.
Common distractors
uses sin or cos instead of tan
Trig confusion under time pressure
A particle in uniform circular motion (period T, radius R) is subsequently launched vertically upward with the SAME speed (v = 2πR/T); find max height. H = v²/(2g) = (2πR/T)²/(2g).
Common distractors
uses radius as height
Surface confusion of geometric R and projectile H
Given launch speed v₀ and angle θ above horizontal, find maximum height. H = v₀² sin² θ / (2g). Common values plug in cleanly when sin θ ∈ {0.5, 0.707, 0.866}. Distractors test (i) using sin instead of sin², (ii) forgetting the factor of 2 in denominator, (iii) using g = 10 vs g = 9.8 inconsistently.
Common distractors
uses sin instead of sin squared
Confusing the height formula H = v²sin²θ/(2g) with the range 2v sin θ / g
Projectile launched at angle θ; find SPEED at highest point. The vertical component vanishes at the apex; horizontal component v₀ cos θ is conserved. Common gotcha: 'angle with the vertical' vs 'angle with the horizontal'.
Common distractors
uses sin instead of cos
Sign-of-angle-axis confusion
speed zero at top
True only for vertical motion (no horizontal v₀)
Buses leave both ends of a route every T minutes; observer travels between them at constant speed; given separate periods at which buses pass observer in same and opposite directions, find bus speed or T. Use relative-velocity equations: same direction T_same = L / (v_bus - v_obs), opposite T_opp = L / (v_bus + v_obs).
Common distractors
ignores direction relative to observer
Treating bus-passing time as absolute
Particle in uniform circular motion at constant speed; identify which of speed / velocity / acceleration / kinetic energy is constant. Speed and KE constant; velocity (vector) and acceleration (centripetal) are NOT constant — direction changes.
Common distractors
treats velocity as scalar
Conflates speed (scalar) with velocity (vector)
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.
Common distractors
drops initial velocity term
Student conflates 'thrown' with 'dropped' and uses v² = 2gh
Sources
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