Kinematics – JEE Main Physics Formula Sheet & Class 11 Notes | Equations of Motion, Projectile, Relative Velocity, Circular Motion & All Kinematics Formulas

Distance, Displacement, Speed, Velocity, Acceleration — Definitions and Calculus Forms

Why the General Calculus Equations of Motion Are the Foundation of All Kinematics JEE Main Questions

Distance and Displacement (from Aakash PDF — Kinematics):

Distance = total length of the actual path traversed between initial and final positions. Always positive, never decreases. SI unit: metre (m). Dimensional formula: [M⁰L¹T⁰].

Displacement = the change in position vector — a vector from initial to final position. Can be positive, negative, or zero. Magnitude of displacement ≤ distance covered.

Speed and Velocity (from Aakash PDF — Kinematics):

Average speed = total distance / total time (scalar, always ≥ 0).

Average velocity = total displacement / total time (vector, can be zero even if speed ≠ 0).

Instantaneous velocity: v = dx/dt (limit of Δx/Δt as Δt→0).

Instantaneous speed = |v| = |dx/dt| (magnitude of instantaneous velocity).

Acceleration (from Aakash PDF — Kinematics):

a = dv/dt (rate of change of velocity with time).

a = v dv/dx (useful when velocity is given as function of position).

General Equations of Motion — Calculus Forms (from Aakash PDF — Kinematics JEE Main):

v = dx/dt → vdt = dx → ∫vdt = displacement = area under v-t graph

a = dv/dt → adt = dv → ∫adt = change in velocity = area under a-t graph

a = vdv/dx → adx = vdv → ∫adx = area under a-x graph = ½(v²–u²)

These three integral forms are the most general — they work even when acceleration is NOT constant.

Equations of Motion — Uniform Acceleration (from Aakash PDF — Kinematics JEE Main):

When a = constant (uniform acceleration):

1. v = u + at

2. s = ut + ½at² = (u+v)t/2 = vt – ½at²

3. v² = u² + 2as

4. Sₙth = u + a(2n–1)/2 (displacement in the nth second specifically)

5. x = x₀ + ut + ½at² (position at time t, where x₀ = initial position)

Here: u = initial velocity (at t=0), v = velocity at time t, a = uniform acceleration, s = displacement during 0 to t, Sₙth = displacement in the nth second (NOT total displacement up to nth second). Download the Free PDF for Kinematics for all equation of motion examples for JEE Main.

Graphs in Kinematics — x-t, v-t, a-t and a-x Graphs

Why Graph Interpretation Is the Most-Tested Kinematics Topic in JEE Main Physics

Position-Time (x-t) Graph (from Aakash PDF — Kinematics JEE Main):

Slope of tangent at any point on x-t graph = instantaneous velocity at that point. Steep positive slope → large positive velocity. Horizontal line → zero velocity (at rest). Negative slope → motion in negative direction.

Slope of chord joining two points on x-t graph = average velocity over that time interval.

x-t graph CANNOT be symmetric about the time-axis — a particle cannot be at two positions at the same instant.

Distance-time graph is always an increasing curve (distance never decreases). Displacement-time graph does NOT show the trajectory of the particle.

Velocity-Time (v-t) Graph (from Aakash PDF — Kinematics JEE Main):

Slope of tangent at any point on v-t graph = instantaneous acceleration at that point.

Slope of chord between two points on v-t graph = average acceleration in that interval.

Area under the speed-t graph between tᵢ and tᶠ = distance covered in interval (tᶠ–tᵢ).

Area under the velocity-t graph (taking sign into account): A₁+A₂ (above axis positive, below negative) = displacement. |A₁|+|A₂| = total distance. Specifically: shaded area (A₁–A₂) = displacement; A₁+A₂ = distance (for v-t graph with area A₁ above axis and A₂ below).

Acceleration-Time (a-t) Graph (from Aakash PDF — Kinematics JEE Main):

Area under a-t graph between tᵢ and tᶠ = change in velocity (vᶠ–vᵢ) in that interval.

Velocity-Position (v-x) Graph (from Aakash PDF — Kinematics JEE Main):

Acceleration of particle at any position x₀: a = v × (dv/dx) = v × (slope of v-x graph at x₀).

Graphically: a = v₀ × tanβ where β is the angle of tangent to the v-x curve at x₀ and v₀ is the speed at x₀.

Key Application — Body starts from rest, accelerates then decelerates (from Aakash PDF — Kinematics):

If a body starts from rest with acceleration α, then decelerates with retardation β and comes to rest, and total time = T:

(a) Maximum velocity: vₘₐₓ = αβT/(α+β)

(b) Length of journey: L = αβT²/[2(α+β)]

(c) Average velocity = vₘₐₓ/2

(d) t₁ = βT/(α+β); t₂ = αT/(α+β); t₁/t₂ = β/α

Velocity at midpoint of path (from Aakash PDF — Kinematics JEE Main):

If particle moves with uniform acceleration on a straight line with velocity vA at A and vB at B, then velocity at the midpoint of AB:

v_mid = √[(vA²+vB²)/2]

This is the root-mean-square of the two end velocities. Download the Free PDF for Kinematics for all graph examples for JEE Main.

Motion Under Gravity — All Cases

Why All 5 Cases of Motion Under Gravity Are Direct Kinematics JEE Main Formulas

Assumption (from Aakash PDF — Kinematics): If the height of the object is small compared to the radius of Earth, motion is uniformly accelerated with g = 9.8 m/s² ≈ 10 m/s² downward. Sign convention: take upward as positive, downward as negative. a = –g always.

Case 1 — Object released from height h (from Aakash PDF — Kinematics JEE Main):

u = 0 (released from rest), taking downward as positive: a = +g.

Time to reach ground: T = √(2h/g)

Velocity when reaching ground: v = √(2gh)

Taking upward as positive: h = 0 – ½gT² → T = √(2h/g); v = –√(2gh) (downward direction).

Case 2 — Particle projected vertically upward from ground with speed u (from Aakash PDF — Kinematics JEE Main):

(a) Time of ascent = Time of descent = T/2 = u/g. Total time of flight T = 2u/g.

(b) Maximum height: H = u²/2g

(c) Speed when it hits the ground = u (symmetric, same as initial speed).

(d) Displacement in complete journey = 0 (returns to start); Distance = 2H = u²/g.

(e) Average velocity = 0; Average speed = u/2.

Special property from PDF: A body takes t seconds to reach highest point. Then: distance in tth second = distance in (t+1)th second; distance in (t–1)th second = distance in (t+2)th second; distance in (t–r)th second = distance in (t+r+1)th second.

Case 3 — Body projected upward from height h with initial speed u (from Aakash PDF — Kinematics JEE Main):

(a) Speed at same level on the way down = u (symmetric about highest point).

(b) Speed at ground level: v = √(u²+2gh)

(c) Time to reach same level: T = 2u/g

(d) Total time of flight T': solve –h = uT' – ½gT'² → T' = (u+√(u²+2gh))/g

Case 4 — Body projected downward from height h with initial speed u (from Aakash PDF — Kinematics JEE Main):

(a) Speed at ground: v = √(u²+2gh)

(b) Time: T = (–u+√(u²+2gh))/g [taking downward positive: h = uT+½gT², solve for T]

Case 5 — Symmetry of distances in nth second for vertical motion (from Aakash PDF — Kinematics):

If maximum height is reached at t = T, the body covers equal distances in symmetric seconds: Sₜ = Sₜ₊₁; Sₜ₋₁ = Sₜ₊₂; Sₜ₋ᵣ = Sₜ₊ᵣ₊₁. Download the Free PDF for Kinematics for all motion under gravity examples for JEE Main.

Vectors — Parallelogram Law, Dot Product, Cross Product, Resolution

Why Vector Operations Are the Essential Mathematical Tool Behind All Kinematics JEE Main Formulas

Parallelogram Law of Vector Addition (from Aakash PDF — Kinematics JEE Main):

If two vectors A and B having a common origin are represented as two adjacent sides of a parallelogram, the diagonal from the common origin represents their resultant R.

|R| = √(A²+B²+2AB cosθ) where θ = angle between A and B.

Direction: tanβ = B sinθ/(A+B cosθ) [angle of R with A]; tan(β') = A sinθ/(B+A cosθ) [angle of R with B].

If |A| = |B| = x: R = 2x cos(θ/2), and R bisects the angle between A and B.

Important special cases (from Aakash PDF — Kinematics JEE Main):

R_max = A+B when θ=0° (vectors parallel). R_min = |A–B| when θ=180° (antiparallel).

R² = A²+B² when θ=90° (vectors perpendicular).

If |A|=|B|=|R|, then θ=120° (each separated by 120°).

If R⊥A: cos θ = –A/B, and A²+R² = B².

For n coplanar vectors of equal magnitude acting at a point with equal angles between consecutive vectors (= 360°/n): resultant = 0.

Vector Subtraction (from Aakash PDF — Kinematics):

A–B = A+(–B). Using parallelogram with angle (π–θ):

|A–B| = √(A²+B²–2AB cosθ)

If |A|=|B|=x: |A–B| = 2x sin(θ/2).

Resolution of Vectors (from Aakash PDF — Kinematics JEE Main):

Any vector V can be resolved into two perpendicular components along x and y axes:

Vx = V cosθ; Vy = V sinθ; V = Vxî+Vyĵ; |V| = √(Vx²+Vy²)

Unit vector along V: V̂ = cosθ î + sinθ ĵ

Dot Product (Scalar Product) (from Aakash PDF — Kinematics JEE Main):

A·B = AB cosθ. If θ<90°: A·B>0. If θ=90°: A·B=0. If θ>90°: A·B<0.

î·î = ĵ·ĵ = k̂·k̂ = 1; î·ĵ = ĵ·k̂ = k̂·î = 0.

A·A = A²; Projection of A on B = (A·B)/|B|; Commutative: A·B = B·A; Distributive: A·(B+C) = A·B+A·C.

Cross Product (Vector Product) (from Aakash PDF — Kinematics JEE Main):

A×B = AB sinθ n̂ where n̂ is the unit vector perpendicular to both A and B (right-hand screw rule).

|A×B| = AB sinθ. If A||B: A×B = 0. î×ĵ = k̂; ĵ×k̂ = î; k̂×î = ĵ (cyclic); ĵ×î = –k̂, etc.

Anti-commutative: A×B = –(B×A). Download the Free PDF for Kinematics for all vector formula examples for JEE Main.

Relative Motion — Relative Velocity, River Crossing, Umbrella, Closest Approach

Why Relative Velocity and River Crossing Are the Most Conceptually Tested Kinematics Topics in JEE Main

Relative Velocity (from Aakash PDF — Kinematics JEE Main):

Velocity of object A with respect to object B: v_AB = v_A – v_B

Velocity of object B with respect to object A: v_BA = v_B – v_A = –v_AB

|v_AB| for two objects moving at angle θ between their velocities: |v_AB| = √(vA²+vB²–2vAvB cosθ)

Direction of Umbrella (from Aakash PDF — Kinematics JEE Main):

A person moving with speed v_M needs to hold umbrella opposite to the direction of relative velocity of rain with respect to man (v_RM = v_R – v_M).

tanθ = v_M/v_R where θ is the angle with vertical that the umbrella should be tilted in the forward direction of motion.

Closest Approach (from Aakash PDF — Kinematics JEE Main):

Two objects A and B at separation x, moving with velocities v_A and v_B. v_AB = v_A–v_B is the relative velocity of A with respect to B. Perpendicular distance from B to the line of v_AB = y (the closest approach distance).

y = x sinθ where sinθ = v_A/|v_AB| (when v_A⊥v_B)

For the case v_A⊥v_B: y = x·v_A/√(v_A²+v_B²) = x tanα/√(1+tan²α) where tanα = v_A/v_B.

River Crossing (from Aakash PDF — Kinematics JEE Main):

Setup: river of width d, river flows with speed u (along river). Man swims with speed v (w.r.t. water). Man swims at angle θ to normal to bank: vx = u–v sinθ (net velocity along river, drift direction), vy = v cosθ (velocity across river).

Time to cross river: t = d/vy = d/(v cosθ)

Drift (distance swept along river): D = vx × t = d(u–v sinθ)/(v cosθ)

Case I — Minimum Time (from Aakash PDF — Kinematics):

Time is minimum when cosθ = 1, i.e., θ = 0° (swim straight across, perpendicular to bank).

Minimum time: t_min = d/v

Drift in this case: D = du/v (man is swept downstream by the river).

Case II — Minimum Distance / Shortest Path (from Aakash PDF — Kinematics):

To cross with zero drift (D=0): u–v sinθ = 0 → sinθ = u/v (only possible if v > u).

Man must swim upstream at angle θ to the normal to compensate for river current.

Time for shortest path: t = d/(v cosθ) = d/√(v²–u²)

If v < u: it is impossible to go straight across; man will always have drift. Download the Free PDF for Kinematics for all relative motion examples for JEE Main.

Download Free PDF — Kinematics JEE Main Physics Formula Sheet

All Kinematics formulas from the Aakash Rapid Revision PDF: distance and displacement definitions, speed vs velocity, instantaneous v=dx/dt, a=dv/dt, a=v dv/dx, three integral forms (∫vdt=displacement, ∫adt=Δv, ∫adx=½Δv²), all 5 equations of uniform motion (v=u+at, s=ut+½at², v²=u²+2as, Sₙth=u+a(2n–1)/2, x=x₀+ut+½at²), all graph properties (x-t slope=v, v-t slope=a, v-t area=displacement, speed-t area=distance, a-t area=Δv, v-x: a=v dv/dx), x-t cannot be symmetric about time-axis, velocity at midpoint v=√[(vA²+vB²)/2], start-stop trip (vmax=αβT/(α+β), L=αβT²/2(α+β)), all 5 gravity cases (free fall, vertical up, projection from height up, projection from height down, symmetry of nth-second distances), all gravity results (T=√(2h/g), v=√(2gh), T_flight=2u/g, H=u²/2g, v=√(u²+2gh)), parallelogram law |R|=√(A²+B²+2ABcosθ), direction formula, all special cases (Rmax, Rmin, R⊥, θ=120°, R⊥A, n coplanar=0), vector subtraction, resolution (Vx=Vcosθ, Vy=Vsinθ), dot product (A·B=ABcosθ, perpendicular→0, î·î=1, î·ĵ=0), cross product (A×B=ABsinθn̂, parallel→0, î×ĵ=k̂), relative velocity (v_AB=v_A–v_B), umbrella (tanθ=v_M/v_R), closest approach (y=x sinθ), river crossing (minimum time θ=0°: t=d/v, D=du/v; minimum distance sinθ=u/v: t=d/√(v²–u²)), horizontal projection (T=√(2H/g), R=u√(2H/g), trajectory gx²/2u²), oblique projection (T=2u sinθ/g, H=u²sin²θ/2g, R=u²sin2θ/g, trajectory x tanθ–gx²/2u²cos²θ, tanθ=gT²/2R=4H/R, Rmax=u²/g at 45°, Rmax/4=H, same R for θ and 90°–θ, R=½gT₁T₂, R=4√(H₁H₂), v_perp at t=u/g sinθ), angular velocity ω=dθ/dt, v=ωr, ac=v²/r=ω²r, aT=αr, net a=√(ac²+aT²), uniform (aT=0), angular kinematic equations.

Why Kinematics Is the Foundation Chapter for All JEE Main Physics

The 4 equations of uniformly accelerated motion (v=u+at, s=ut+½at², v²=u²+2as, Sₙth) cover 80% of Kinematics JEE Main questions. Every problem involving constant acceleration — free fall, horizontal projection, motion on inclined plane, rocket launch — uses these four equations. The key skill is choosing the right sign convention (usually upward positive), identifying u, a, t, s correctly, and substituting. The nth-second formula Sₙth = u+a(2n–1)/2 is the most frequently misunderstood — it gives displacement in the nth second specifically, not total displacement up to nth second.

Projectile motion is a superposition problem — treat horizontal and vertical components completely independently. Horizontal: uniform motion, no acceleration, uₓ = u cosθ constant throughout. Vertical: uniformly accelerated motion with a = –g. The trajectory equation y = x tanθ(1–x/R) follows immediately by eliminating t from the two parametric equations. Every projectile question in JEE Main either asks for T, H, R directly; tests the complementary angle property; or tests the velocity direction at a specific instant.

River crossing is a 2-minute guaranteed score in Kinematics JEE Main. Two cases — minimum time (swim perpendicular, θ=0°) and minimum distance (swim upstream at sinθ=u/v) — need to be memorised with their respective time and drift formulas. The only tricky case: if river speed u > man's speed v, zero drift is impossible — the river will always sweep him downstream.

Graph problems test conceptual understanding of area = displacement and slope = velocity/acceleration. The most common JEE Main graph trap: area under v-t graph = displacement (algebraic, taking sign), not distance. Distance = area under speed-t graph (all positive). And: slope of x-t = velocity (not speed), so a negative slope means motion in the negative direction. Download the Free PDF for Kinematics to have all formulas ready.

Who Should Use This Kinematics Formula Sheet?

Learning Outcomes After Completing Kinematics

After working through Kinematics using this formula sheet, a student should confidently accomplish the following for JEE Main physics. On basic motion: apply v=dx/dt and a=dv/dt for any motion; use the third kinematic form a=v dv/dx; compute average velocity and average speed correctly; apply all 5 equations of uniform motion (v=u+at, s=ut+½at², v²=u²+2as, Sₙth, x=x₀+ut+½at²); use the nth-second formula correctly.

On graphs: read slope of x-t to get instantaneous velocity; read area under v-t to get displacement (with sign) and distance (without sign); read slope of v-t to get acceleration; read area under a-t to get change in velocity; use v-x graph slope to compute acceleration; interpret multi-phase v-t graphs; apply velocity-at-midpoint formula v=√[(vA²+vB²)/2].

On gravity: apply all 5 gravity cases correctly; use symmetry of vertical motion (tₐₛ=tdes, Sₙth = S(n+1)th for appropriate n, etc.); find time of flight and landing speed for all projection directions.

On vectors: compute resultant using parallelogram law; find direction; handle all special cases (perpendicular, parallel, antiparallel, n equal vectors); resolve into components; compute dot and cross products including in component form.

On relative motion: compute v_AB = v_A–v_B; find umbrella angle; find closest approach; solve both river crossing cases (minimum time and minimum distance) including when the crossing angle and drift.

On projectile: apply all formulas (T, H, R, trajectory) for both horizontal and oblique projection; use complementary angle property; use tanθ=gT²/2R and tanθ=4H/R to find angle; use R=½gT₁T₂ and R=4√(H₁H₂) for two-angle problems; find velocity at any instant including when perpendicular to initial.

On circular motion: relate ω, v, r, T, f; compute centripetal ac=v²/r and tangential aT=αr accelerations; compute net acceleration in non-uniform circular motion; apply angular kinematic equations. Download the Free PDF for Kinematics to test all outcomes before your JEE Main exam.

Get the Free PDF for Kinematics — JEE Main Quick Revision

The Aakash Rapid Revision & Formula Bank PDF for Kinematics contains all equations of motion, all graph rules, all gravity cases, all vector operations, all relative motion results, all projectile formulas, and all circular motion results in one structured JEE Main physics reference.

Conclusion — Kinematics: The Motion Language of JEE Main Physics

Kinematics is the chapter that teaches you the formal language of motion — displacement, velocity, acceleration, and their relationships through calculus and algebraic equations. Every physics chapter from Laws of Motion to Electromagnetism uses kinematic language. Mastering Kinematics means you can set up any motion problem correctly, choose the right equations, and solve without confusion about signs or directions.

The three pillars of Kinematics for JEE Main: (1) Equations of motion — memorise all 5 forms, especially Sₙth; (2) Projectile motion — four formulas (T, H, R, trajectory) are the complete toolbox; (3) Relative motion and graphs — these are the conceptual tests that separate 70-percentile students from 99-percentile. Use this page and the Free PDF Download for Kinematics as your complete JEE Main revision foundation.

Frequently Asked Questions — Kinematics Formulas

Related Formula Sheets — JEE Main Physics

This is a free educational resource for JEE Main 2025–2026, JEE Advanced, NEET, Class 11 CBSE, and BITSAT Physics preparation. The PDF formula sheet for Kinematics is sourced from Aakash's Rapid Revision & Formula Bank. All Kinematics formulas and content are for academic reference only.