Class 11 · Chapter 3

Kinematics

Overview, notes, short notes, formula sheet, daily practice problems, previous year questions, and videos for this chapter — all in one place.

Kinematics Formula Sheet

20 formulas across 6 topics in Kinematics.

3 min read

Updated 2026-07-02 · v1.0.1

Basic Definitions

Every kinematics problem starts here — distance vs displacement, speed vs velocity, and what acceleration really measures.

v̄ = Δx / Δt

Average velocity

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v̄	average velocity	m/s	[LT⁻¹]
Δx	displacement	m	[L]
Δt	time interval	s	[T]

Valid when

Uses displacement, not distance

Common mistakes

Using total distance instead of displacement — that gives average speed, not average velocity

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Average speed = Total distance / Total time

Average speed

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v̄	average speed	m/s	[LT⁻¹]

Common mistakes

For a two-part journey at speeds v₁ and v₂ over equal distances, average speed is the harmonic mean 2v₁v₂/(v₁+v₂), NOT the arithmetic mean

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v = dx/dt

Instantaneous velocity

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Limit of average velocity as Δt → 0

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v	instantaneous velocity	m/s	[LT⁻¹]
x	position	m	[L]
t	time	s	[T]

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a = dv/dt = v (dv/dx)

Instantaneous acceleration (both forms)

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Chain rule: dv/dt = (dv/dx)(dx/dt) = v·dv/dx

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
a	acceleration	m/s²	[LT⁻²]
v	velocity	m/s	[LT⁻¹]

Valid when

The v·dv/dx form is for when velocity is given as a function of position

Common mistakes

Forgetting the v·dv/dx form — JEE loves giving v(x) and asking for acceleration

Read the concept →

Worth remembering

Distance ≥ |displacement| always; equality holds only for straight-line motion without reversal
A body can have zero velocity and non-zero acceleration at the same instant (top of vertical throw)

Equations of Motion (Constant Acceleration)

The three SUVAT equations — valid ONLY when acceleration is constant. This is the most-used formula set in all of Class 11 physics.

v = u + at

First equation of motion

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Integrating a = dv/dt with constant a

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v	final velocity	m/s	[LT⁻¹]
u	initial velocity	m/s	[LT⁻¹]
a	acceleration	m/s²	[LT⁻²]
t	time	s	[T]

Valid when

Constant acceleration only
Straight-line motion (or apply per-component)

Common mistakes

Using it when acceleration varies with time — integrate instead
Sign errors: choose a positive direction and stick to it for u, v, a

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s = ut + ½at²

Second equation of motion

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Integrating v = dx/dt after substituting v = u + at

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
s	displacement	m	[L]
u	initial velocity	m/s	[LT⁻¹]
a	acceleration	m/s²	[LT⁻²]
t	time	s	[T]

Valid when

Constant acceleration only

Common mistakes

s is displacement from the STARTING point, not total distance if direction reverses

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v² = u² + 2as

Third equation of motion

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Eliminating t between the first two equations (or from v·dv/dx = a)

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v	final velocity	m/s	[LT⁻¹]
u	initial velocity	m/s	[LT⁻¹]
a	acceleration	m/s²	[LT⁻²]
s	displacement	m	[L]

Valid when

Constant acceleration only
Time-independent — use when t is not given/needed

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sₙ = u + a(2n−1)/2

Displacement in the nth second

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s(n) − s(n−1) using the second equation

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
sₙ	displacement in the nth secondnumerically equals metres covered during that one second	m	[L]
n	second number (1st, 2nd, ...)	dimensionless	[M⁰L⁰T⁰]

Valid when

Constant acceleration only

Common mistakes

Dimensional panic: the formula looks dimensionally odd but is correct — it is displacement per unit (1 s) interval

Read the concept →

Worth remembering

For free fall take a = +g downward if you choose down as positive — consistency beats convention
Under gravity alone, time up = time down and speed of projection = speed of return (no air resistance)

NEET asks these directly; JEE Main wraps them in graphs; JEE Advanced hides them inside multi-body constraint problems.

Motion Under Gravity (Free Fall)

Equations of motion with a = g ≈ 9.8 m/s², taking consistent sign convention.

H = u²/2g

Maximum height (vertical throw)

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Third equation with v = 0 at the top

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
H	maximum height	m	[L]
u	initial upward speed	m/s	[LT⁻¹]
g	acceleration due to gravity	m/s²	[LT⁻²]

Valid when

No air resistance

Read the concept →

T = 2u/g

Total time of flight (vertical throw, return to launch point)

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Time up (u/g) doubled by symmetry

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T	total time of flight	s	[T]
u	initial upward speed	m/s	[LT⁻¹]
g	acceleration due to gravity	m/s²	[LT⁻²]

Valid when

Returns to the same height
No air resistance

Read the concept →

Worth remembering

Velocity at any height h on the way up equals velocity at the same h on the way down (magnitude)
Galileo's odd-number rule: distances in successive seconds of free fall from rest are in ratio 1:3:5:7...

Projectile Motion

Two independent motions: uniform velocity horizontally, uniform acceleration (g) vertically. Every projectile result comes from treating them separately.

T = (2u sinθ) / g

Time of flight

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Vertical motion: uᵧ = u·sinθ, returns when vertical displacement = 0

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
T	time of flight	s	[T]
u	launch speed	m/s	[LT⁻¹]
θ	launch angle from horizontal	rad (or °)	[M⁰L⁰T⁰]
g	acceleration due to gravity	m/s²	[LT⁻²]

Valid when

Level ground (launch and landing at same height)
No air resistance

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H = (u² sin²θ) / 2g

Maximum height

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Third equation applied to vertical component with vᵧ = 0 at top

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
H	maximum height	m	[L]
u	launch speed	m/s	[LT⁻¹]
θ	launch angle from horizontal	rad (or °)	[M⁰L⁰T⁰]

Valid when

No air resistance

Read the concept →

R = (u² sin 2θ) / g

Horizontal range

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R = (u·cosθ)·T with T = 2u·sinθ/g and identity 2sinθcosθ = sin2θ

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
R	horizontal range	m	[L]
u	launch speed	m/s	[LT⁻¹]
θ	launch angle from horizontal	rad (or °)	[M⁰L⁰T⁰]

Valid when

Level ground
No air resistance

Common mistakes

Range is maximum at θ = 45°, and R(θ) = R(90°−θ): complementary angles give equal ranges
sin2θ means sin(2θ), not (sinθ)² — read carefully under exam pressure

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y = x tanθ − gx² / (2u² cos²θ)

Equation of trajectory

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Eliminate t between x = u·cosθ·t and y = u·sinθ·t − ½gt²

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
y	height at horizontal distance x	m	[L]
x	horizontal distance	m	[L]

Valid when

No air resistance

Common mistakes

Alternate form y = x·tanθ(1 − x/R) is faster when range R is known

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t = √(2h/g)

Fall time — horizontal projectile from height h

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Vertical motion from rest: h = ½gt²

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
t	time to reach ground	s	[T]
h	initial height	m	[L]

Valid when

Launched horizontally (uᵧ = 0)
No air resistance

Common mistakes

Fall time is independent of the horizontal launch speed — a dropped ball and a horizontally-thrown ball land together

Read the concept →

Worth remembering

At maximum height, velocity is purely horizontal: v = u·cosθ (never zero unless θ = 90°)
Kinetic energy at top = (initial KE)·cos²θ — a favourite one-liner in NEET

JEE Advanced frequently launches projectiles from moving platforms or inclines — the level-ground formulas above then need re-derivation from components.

Relative Motion

Velocity of A as seen from B. One vector subtraction solves rain-man, river-boat, and chase problems.

v(AB) = v(A) − v(B)

Relative velocity of A with respect to B

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v(AB)	velocity of A relative to B	m/s	[LT⁻¹]
v(A)	velocity of A (ground frame)	m/s	[LT⁻¹]
v(B)	velocity of B (ground frame)	m/s	[LT⁻¹]

Common mistakes

This is a VECTOR subtraction — for opposite directions magnitudes add, for same direction they subtract

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sin α = vᵣ / vᵇ (upstream tilt for straight crossing)

River crossing — shortest path condition

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Boat's upstream velocity component must cancel the river flow

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
α	angle of boat heading upstream from the perpendicular	rad (or °)	[M⁰L⁰T⁰]
vᵣ	river flow speed	m/s	[LT⁻¹]
vᵇ	boat speed in still water	m/s	[LT⁻¹]

Valid when

Only possible when vᵇ > vᵣ

Common mistakes

Shortest PATH needs the upstream tilt; shortest TIME needs heading straight across (perpendicular) — students swap these constantly

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d(min) = d sinθ, θ between the line joining them and v(rel)

Minimum distance of approach (relative motion)

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In the frame of one body, the other moves in a straight line along v(rel); the closest approach is the perpendicular from the stationary body to that line

Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
d	initial separation between the two bodies	m	[L]
θ	angle between the initial line joining them and the relative velocity	rad (or °)	[M⁰L⁰T⁰]
v(rel)	relative velocity (constant)	m/s	[LT⁻¹]

Valid when

Both bodies move with constant velocities
Separation is minimum when the relative velocity is perpendicular to the line joining them

Common mistakes

Working in the ground frame with two moving bodies — switch to the frame of one body and the problem becomes a point and a straight line

Read the concept →

Worth remembering

Rain-man problems: tanθ of the tilted umbrella comes from the vector triangle of the rain velocity and the man's velocity

Graphical Analysis

What slopes and areas mean on each graph — JEE Main's favourite way to disguise simple kinematics.

slope of x–t graph = velocity

Position–time graph slope

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
x	position	m	[L]
t	time	s	[T]

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slope of v–t = acceleration; area under v–t = displacement

Velocity–time graph: slope and area

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Variables used in this formula, with units and dimensions
Symbol	Meaning	Unit	Dimension
v	velocity	m/s	[LT⁻¹]
t	time	s	[T]

Common mistakes

Area BELOW the time axis is negative displacement; take |areas| and add for total distance

Read the concept →

Worth remembering

An x–t graph can never be a vertical line or double back — that would mean two positions at one time
Uniform acceleration ⇒ straight-line v–t graph and parabolic x–t graph

When a graph question appears, first write what slope and area mean for THAT graph — half the question solves itself.

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