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. 5Position Time Graph
Lesson
Position-time graphs encode an object's entire motion history in a single picture. The one trap that costs marks on this topic: confusing trig functions when extracting velocity from the graph's angle.
A position-time (x-t) graph plots position on the vertical axis and time on the horizontal axis (NCERT Class 11 Physics Chapter 2, page 5). Each point on the curve tells you where the object is at that instant. The slope of the graph at any point gives the instantaneous velocity.
For a straight-line x-t graph, the slope is constant — the object moves with uniform velocity. The velocity equals the tangent of the angle the line makes with the time axis:
v = slope = tan θ (where θ is measured from the t-axis)
Here is the high-frequency trap: when a question gives you the angle, you must use tan, not sin or cos. Under exam pressure, students reach for sin θ or cos θ out of reflex. The slope of any graph is rise/run = Δx/Δt, and the trigonometric ratio that equals opposite/adjacent for an angle in a right triangle is the tangent.
A horizontal line on an x-t graph means zero slope — the object is at rest. A steeper line means higher velocity. If two lines have different angles (say 30° and 60°), the velocity ratio is tan 30° : tan 60° = (1/√3) : √3 = 1 : 3.
For a curved x-t graph, the slope changes with time. The tangent drawn at any point gives the instantaneous velocity at that moment. A parabolic x-t curve (x = x₀ + v₀t + ½at²) indicates uniformly accelerated motion — the slope increases (or decreases) steadily.
Watch out: the angle the tangent line makes with the t-axis gives velocity via tan, not via sin or cos. This holds for curved graphs too — draw the tangent, measure the angle from the t-axis, and take the tangent of that angle.
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 position-time graph of a particle moving along a straight line is a straight line making an angle of 45° with the time axis. What is the velocity of the particle?
Two particles A and B have straight-line position-time graphs making angles of 30° and 60° respectively with the time axis. What is the ratio of velocity of A to velocity of B?
The position-time graph of an object is a horizontal straight line at x = 5 m. Which statement is correct?
A particle's position-time graph is a straight line passing through the origin with a positive slope. If the angle this line makes with the time axis is θ, which expression gives the position of the particle at time t?
The position-time graph of a moving car is a parabola opening upward. This indicates that the car is:
The x-t graph of particle P is a straight line making 60° with the t-axis, and the x-t graph of particle Q is a straight line making 30° with the t-axis. Both start from the origin. At time t, the ratio of the displacement of P to that of Q is:
The position-time graph of an object consists of two connected straight-line segments: the first segment rises steeply from t = 0 to t = 5 s, and the second segment is horizontal from t = 5 s to t = 10 s. During the interval t = 5 s to t = 10 s, the object:
The position-time graphs of two objects A and B are straight lines that intersect at a point (t₁, x₁). Which statement is necessarily true at t = t₁?
Quick recall before you leave
Worked Example
Pattern: Position-time graph slope — extract velocity ratio from given angles (anchored to PYQ pattern NEET pattern: position time graph slope, observed 2022).
- 1
Given
The position-time graphs of two particles X and Y are straight lines making angles 30° and 45° with the time axis respectively.
- 2
Required
Find the ratio of the velocity of X to the velocity of Y (v_X : v_Y).
- 3
Concept
The velocity of an object is the slope of its position-time graph. For a straight-line x-t graph making angle θ with the time axis, the slope is tan θ.
- 4
Formula
v = tan θ
- 5
Substitution
v_X = tan 30° = 1/√3 v_Y = tan 45° = 1
- 6
Calculation
v_X / v_Y = (1/√3) / 1 = 1/√3 Note on exact values: 30° and 45° are exact angle specifications from the problem. The trigonometric values tan 30° = 1/√3 and tan 45° = 1 are exact mathematical constants. They do not limit significant figures.
- 7
Final answer
v_X : v_Y = 1 : √3
- 8
Common trap
Using sin or cos instead of tan. If a student used sin: sin 30°/sin 45° = 0.5/0.707 ≈ 0.707, giving the wrong ratio 1 : √2. If they used cos: cos 30°/cos 45° = (√3/2)/(1/√2) = √6/2, also wrong. Only tan gives the slope.
- 9
Similar NEET-style question
The x-t graphs of cars A and B are straight lines through the origin, making angles 60° and 30° with the t-axis. Find the ratio of speeds v_A : v_B. Answer: tan 60° / tan 30° = √3 / (1/√3) = 3. So v_A : v_B = 3 : 1. ---
Before solving, remember these
The slope of the position–time (x-t) graph at any instant gives the instantaneous velocity. A straight x-t line implies uniform velocity; a curved x-t line implies non-zero acceleration.
-- NCERT Class 11 Physics, Ch. 2, p. 2Formulas
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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