Kinematics Formula Sheet — JEE Main Physics

Q: What are the most important Kinematics formulas for JEE Main?

This Kinematics formula sheet covers all the high-yield Physics formulas, definitions and theorems you need for JEE Main, across Frame of reference, Motion in a straight line, Speed and velocity, Uniform and non-uniform motion, Average speed and instantaneous velocity — each shown with the key result and, where useful, a worked example.

Q: Is this Kinematics formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

Q: How should I revise Kinematics formulas?

Blurt the Kinematics formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Every key Kinematics formula, definition and theorem for JEE Main Physics in one place — with common examiner traps and worked examples. Free to read; blurt from memory, then check your gaps.

Syllabus — topics coveredNTA · 12 sub-topics

Frame of reference
Motion in a straight line
Speed and velocity
Uniform and non-uniform motion
Average speed and instantaneous velocity
Uniformly accelerated motion
Velocity-time and position-time graphs
Relations for uniformly accelerated motion
Relative velocity
Motion in a plane
Projectile motion
Uniform circular motion

Motion Basics: Distance, Velocity, Acceleration

Kinematics & frame of reference: Describing motion (position vs time) without its cause. A

frame =

origin

+

axes

+

clock; motion is

absolute

(fixed frame) or

relative

(moving frame).

Comparative: scalars vs vectors of motion

Quantity	Type	Sign?	Note
Distance / path	scalar	$\geq 0$	never decreases
Displacement $Δ r$	vector	$\pm$	end-points only
Speed $∣ v ∣$	scalar	$\geq 0$	instant: $= ∣ v ∣$ ; avg speed $\geq ∣ v_{a v} ∣$
Velocity $v$	vector	$\pm$	tangent to path

Distance vs displacement

▸ $Distance$ : total path covered ( $\geq 0$ ). $Displacement$ $Δ r = r_{2} - r_{1}$ (vector).
▸ $∣Δ r ∣ \leq$ distance — equal only for straight-line motion with $no reversal$ .

Average velocity / speed

v_{a v} = \frac{Δ r}{Δ t};

avg speed = \frac{distance}{Δ t}

Δ r =

displacement,

Δ t =

time interval; avg speed

\geq ∣ v_{a v} ∣

;

v_{a v} =

chord slope on x-t

Instantaneous velocity / acceleration

v = \frac{d r}{d t}; a = \frac{d v}{d t} = \frac{d ^{2} r}{d t ^{2}}

v =

tangent slope on x-t;

a =

slope on v-t

🚫 Examiner Trap · Motion basics

(1)

v = 0

does

NOT

mean

a = 0

— at the top of a throw

v = 0

a = g

. (2) Negative velocity means direction,

not

slowing down (slowing

\Leftrightarrow a

opposes

v

). (3) avg velocity

\neq =

avg speed in general; over a closed loop

v_{a v} = 0

but avg speed

\neq = 0

Motion types

▸ $Uniform$ : equal displacements in equal times $\Rightarrow v$ const, $a = 0$ .
▸ $Non-uniform$ : $v$ changes; if $a$ const $\to$ uniformly accelerated.

Variable acceleration (calculus)

Three forms of

a

v = \frac{d x}{d t}; a = \frac{d v}{d t}; a = v \frac{d v}{d x}

use

a = v d v / d x

when a is a function of position x

Method (choose by what

a

depends on)

▸a(t) given: $v = u + \int a d t$ , then $x = x_{0} + \int v d t$ .
▸a(x) given: $\int v d v = \int a d x$ (separate variables — don't integrate in t).
▸a(v) given: $\int \frac{d v}{a} = \int d t$ or $\int \frac{v d v}{a} = \int d x$ .

⚡ Shortcut · Which calculus form?

Need v(x) but only have a? Use

a = v d v / d x

to skip time entirely. Need x(t)? Integrate twice in t. Picking the wrong form wastes the most exam time here.

Equations of Motion & Motion Under Gravity

Comparative: the 4 SUVAT equations

Equation	Missing var	Use when
$v = u + a t$	$s$	no displacement
$s = u t + \frac{1}{2} a t^{2}$	$v$	final speed unknown
$v^{2} = u^{2} + 2 a s$	$t$	time not needed
$s = \frac{u + v}{2} t$	$a$	accn unknown

Symbols (all signed along chosen

+

axis)

▸ $u =$ initial velocity, $v =$ final velocity, $a =$ (constant) acceleration, $s =$ displacement, $t =$ time.

Distance in the

n^{th}

second

S_{n} = u + \frac{1}{2} a (2 n - 1)

S_{n} =

distance covered during the nth second alone (a velocity-like quantity), NOT total distance in n s

🚫 Examiner Trap · Equations of motion

(1) Valid

only

while a is constant in magnitude AND direction — not for a(t). (2) Use

displacement

s in

v^{2} = u^{2} + 2 a s

, not distance. (3)

S_{n}

is per-second distance, not total. (4) Fix

+

direction FIRST and keep u,v,a,s signed throughout.

Motion under gravity (

a = g \approx 9.8 m s^{- 2}

)

Thrown up with speed

u

t_{asc} = \frac{u}{g} = t_{desc};

H_{m a x} = \frac{u ^{2}}{2 g};

T = \frac{2 u}{g}

t_{asc} =

rise time,

H_{m a x} =

peak height,

T =

total flight time; returns with the same speed u

Free fall from rest

v = g t; h = \frac{1}{2} g t^{2}; v = 2 g h

h =

fall height,

v =

speed after falling

h

Stopping distance

d = \frac{u ^{2}}{2 a} \propto u^{2}

a =

retardation; double the speed

\Rightarrow 4 \times

stopping distance

⚡ Shortcut · Galileo's odd-number rule

From rest, distances in successive equal time intervals are

1 : 3 : 5 : 7 : \dots

A body crosses any height twice with equal speed;

t_{1}, t_{2}

are symmetric about the apex and

t_{1} t_{2} = 2 h / g

🚫 Examiner Trap · Gravity sign

Choose UP as

+

\Rightarrow a = - g

throughout (rise and fall). The most common slip is flipping g's sign at the top — don't; only v reverses.

Motion Graphs

x-t parabola for a>0 and line for a=0; v-t lines for a>0, a=0 and a<0, with area = displacement — x-t (parabola if $a \neq = 0$ ) and v-t (area $=$ displacement).

Comparative: what slope & area mean

Graph	Slope gives	Area gives
$x$ – $t$	velocity $v$	— (no meaning)
$v$ – $t$	acceleration $a$	displacement (signed)
$a$ – $t$	jerk $d a / d t$	$Δ v$ (change in velocity)

Position–time (

x

t

)

▸Tangent slope $=$ instantaneous velocity; chord slope $=$ average velocity.
▸Straight $\Rightarrow$ uniform velocity; parabola $\Rightarrow$ uniform accn. Curving up $\Rightarrow a > 0$ , down $\Rightarrow a < 0$ .

Velocity–time (

v

t

)

▸Slope $=$ acceleration. $Signed area =$ displacement; $|area|$ (all $+$ ) $=$ distance.

🚫 Examiner Trap · Reading motion graphs

(1) On v-t, area below the axis is

negative

displacement — for distance, add the

magnitudes

. (2) x-t slope is velocity, NOT acceleration. (3) No graph can be vertical (infinite velocity/accn). (4) A horizontal x-t tangent

\Rightarrow v = 0

(turning point), but a may be

\neq = 0

⚡ Shortcut · Sign clues

v and (v-t slope)

same

sign

\Rightarrow

speeding up;

opposite

\Rightarrow

slowing. The particle reverses direction where the v-t curve crosses the time-axis.

Continuity

▸Position and velocity change continuously — never by a jump (a velocity jump needs an infinite force). So $x$ - $t$ is smooth (no corners) and $v$ - $t$ has no gaps.
▸Acceleration $can$ jump (a force switched on/off): a-t may have steps, and v-t then has a $corner$ there.

Vectors

Scalar vs vector:

Scalar

: magnitude only (distance, speed, mass).

Vector

: magnitude

+

direction (displacement, velocity, force); obeys triangle / parallelogram law.

Parallelogram law: A and B at angle theta, resultant R — Parallelogram law of addition.

Resultant (angle

θ

between

A, B

)

R = A^{2} + B^{2} + 2 A B cos θ

A, B =

magnitudes; direction of R from

A

tan α = \frac{B sin θ}{A + B cos θ}

Special cases & bounds

▸ $θ = 0^{\circ}$ : $R = A + B$ (max). $θ = 18 0^{\circ}$ : $R = ∣ A - B ∣$ (min). $θ = 9 0^{\circ}$ : $R = A^{2} + B^{2}$ .
▸Always $∣ A - B ∣ \leq R \leq A + B$ .

Resolution & components

A = A_{x} \hat{i} + A_{y} \hat{j};

A_{x} = A cos θ,

A_{y} = A sin θ

∣ A ∣ = A_{x}^{2} + A_{y}^{2}

tan θ = A_{y} / A_{x}

; in 3D add

A_{z} \hat{k}

Comparative: dot vs cross product

	Dot $A \cdot B$	Cross $A \times B$
Result	scalar	vector
Value	$A B cos θ$	$A B sin θ \overset{n}{^}$
Max at	$θ = 0^{\circ}$	$θ = 9 0^{\circ}$
Zero at	$θ = 9 0^{\circ}$ ( $⊥$ )	$θ = 0^{\circ}$ (parallel)
Commutative?	yes	no ( $A \times B = - B \times A$ )

Unit vectors

▸ $\hat{i}, \hat{j}, \hat{k}$ mutually $⊥$ , magnitude 1; $\overset{n}{^} = A /∣ A ∣$ . Addition is commutative & associative.

🚫 Examiner Trap · Vectors

(1) Add vectors head-to-tail,

never

add magnitudes (

∣ A + B ∣ \neq = A + B

unless parallel). (2)

\overset{n}{^}

(cross-product direction) by the right-hand rule — order matters. (3) Components use the angle WITH that axis:

A cos θ

along the axis

θ

is measured from.

⚡ Shortcut · Resolved motion

Plane motion

=

two

independent

1-D motions along

⊥

axes; solve x and y separately. This is the entire basis of projectile motion.

Relative Velocity, River–Boat & Rain–Man

Relative velocity

v_{A B} = v_{A} - v_{B}; v_{A B} = - v_{B A}

v_{A B} =

velocity of A as seen from B;

v_{A}, v_{B} =

ground velocities

One dimension

▸Same direction: $v_{A B} = ∣ v_{A} - v_{B} ∣$ . Opposite: $v_{A B} = v_{A} + v_{B}$ (approach/separation rate).

River of width d: swimmer velocity v across, river velocity u added head-to-tail, resultant v+u reaching the far bank downstream by the drift ud/v — Swimmer $v$ (relative to water) $+$ river $u$ $=$ ground velocity; drift $= u d / v$ when heading straight across.

Comparative: river crossing (

d

width,

u

river,

v

swim)

Goal	Heading	Result
Min time	straight across ( $θ = 0$ )	$t = \frac{d}{v}$ , drift $= \frac{u d}{v}$
Zero drift	upstream, $sin θ = \frac{u}{v}$	$t = \frac{d}{v ^{2} - u ^{2}}$
General $θ$	$θ$ to normal	$t = \frac{d}{v cos θ}$

v < u

(slow swimmer)

▸Cannot reach the point directly opposite. Least drift: head upstream at $θ$ to the $normal$ (the perpendicular to the bank) with $sin θ = \frac{v}{u}$ ; drift $_{m i n} = \frac{d u ^{2} - v ^{2}}{v}$ .

Rain v_r down, man v_m horizontal; v_r plus minus v_m head-to-tail gives v_rm, at angle theta to the vertical — Rain relative to man $= v_{r} - v_{m}$ .

🎯 Exam · Rain–man

Hold the umbrella along

v_{r m} = v_{r} - v_{m}

: tilt from vertical by

θ = tan^{- 1} \frac{v _{m}}{v _{r}}

towards

your motion.

v_{r} =

rain speed,

v_{m} =

man's speed.

🚫 Examiner Trap · Relative velocity

(1) Order matters:

v_{A B} = v_{A} - v_{B}

(the second subscript is the observer). (2) Min-time and min-drift headings are

different

— min time means head straight across (accept drift). (3) Tilt the umbrella INTO the relative-velocity direction, toward your motion, not away.

Same idea, other set-ups

▸ $Aeroplane in wind$ $=$ boat in river (air is the river). To fly due N in an E-wind w at airspeed v: head W of N with $sin θ = \frac{w}{v}$ ; ground speed $v^{2} - w^{2}$ .
▸ $Ball in a lift$ accelerating up at a: relative to the lift use $g_{eff} = g + a$ (down at a: $g - a$ ; free fall: $0$ ).
▸ $Two projectiles$ in flight: relative acceleration $= g - g = 0$ , so each sees the other move in a $straight line$ at constant velocity; they collide only if $v_{rel}$ points along the line joining them.
▸ $Closest approach$ (constant $v_{rel}$ , starting separation $r_{0}$ ): $d_{m i n} = \frac{∣ r _{0} \times v _{rel} ∣}{∣ v _{rel} ∣}$ at $t = - \frac{r _{0} \cdot v _{rel}}{∣ v _{rel} ∣ ^{2}}$ .

Projectile Motion

Parabolic projectile: u at angle theta, height H, range R, vx at top — Parabolic path; $v_{x} = u cos θ$ constant.

Components (

a_{x} = 0, a_{y} = - g

)

v_{x} = u cos θ;

v_{y} = u sin θ - g t

u =

launch speed,

θ =

launch angle from horizontal; x-motion uniform, y-motion uniformly accelerated

Trajectory (parabola)

y = x tan θ - \frac{g x ^{2}}{2 u ^{2} cos ^{2} θ}

= x tan θ (1 - \frac{x}{R})

Flight time / Height / Range

T = \frac{2 u sin θ}{g};

H = \frac{u ^{2} sin ^{2} θ}{2 g};

R = \frac{u ^{2} sin 2 θ}{g}

T =

time of flight,

H =

max height,

R =

horizontal range; also

R = 4 H / tan θ

Key facts

▸R is maximum at $θ = 4 5^{\circ}$ : $R_{m a x} = u^{2} / g$ . Same R for $θ$ and $9 0^{\circ} - θ$ (complementary).
▸At the top $v_{y} = 0$ , speed $= u cos θ$ (NOT zero); $T = 2 t_{asc}$ .

Comparative: launch types

Launch	Flight time	Range
Ground, angle $θ$	$\frac{2 u sin θ}{g}$	$\frac{u ^{2} sin 2 θ}{g}$
Horizontal from $h$	$\frac{2 h}{g}$	$u \frac{2 h}{g}$
Up incline $α$	$\frac{2 u sin θ}{g cos α}$	see below

Speed & direction at time

t

v = u^{2} - 2 u sin θ g t + g^{2} t^{2};

tan ϕ = \frac{u sin θ - g t}{u cos θ}

ϕ =

angle of the velocity with the horizontal at time t

Projectile launched at angle theta to an incline of angle alpha; range R measured along the incline — $θ$ from the incline, $α$ = incline angle, R along the incline.

Incline (launch

θ

to incline of angle

α

)

R = \frac{u ^{2}}{g cos ^{2} α}

[sin (2 θ + α) - sin α];

R_{m a x}^{↑} = \frac{u ^{2}}{g ( 1 + sin α )}

max range up the plane at

θ = \frac{π}{4} - \frac{α}{2}

; down the plane:

R_{m a x} = \frac{u ^{2}}{g ( 1 - sin α )}

🚫 Examiner Trap · Projectile motion

(1) At the top the speed is

u cos θ

not zero

— only

v_{y} = 0

. (2) g acts at every point, including the top. (3) Horizontal range needs

sin 2 θ

(so

θ = 4 5^{\circ}

max), not

sin θ

. (4) For horizontal projection, time depends

only

on height h, not on u.

Circular Motion (Kinematics)

Uniform circular motion: Motion on a circle of radius r at constant speed v. Speed is constant but

v

(direction) changes

\Rightarrow

the body

is accelerated

(centripetal).

Angular velocity & speed

ω = \frac{d θ}{d t}; v = r ω

ω =

angular velocity (rad/s),

r =

radius;

ω

is the same for every point of a rigid body

Centripetal acceleration

a_{c} = \frac{v ^{2}}{r} = ω^{2} r = 4 π^{2} ν^{2} r

directed radially

inward

(toward centre);

ν =

frequency

Period & frequency

T = \frac{2 π}{ω} = \frac{2 π r}{v};

ν = \frac{1}{T};

v = 2 π r ν

T =

period (s),

ν =

frequency (Hz)

Comparative: centripetal vs tangential accn

	Centripetal $a_{c}$	Tangential $a_{t}$
Direction	radially inward	along tangent ( $∥ v$ )
Formula	$v^{2} / r = ω^{2} r$	$d v / d t = r α$
Changes	direction of $v$	magnitude (speed)
Uniform CM	present	zero

Non-uniform: net acceleration

a = a_{c}^{2} + a_{t}^{2};

tan ϕ = \frac{a _{t}}{a _{c}}

α =

angular acceleration;

ϕ =

angle of net

a

from the radius

🚫 Examiner Trap · Circular motion

(1) Uniform circular motion IS accelerated — 'constant speed' fools students into writing

a = 0

. (2)

a_{c}

points INWARD, not outward (no real 'centrifugal' force in an inertial frame). (3) Here only kinematics: centripetal

force

m v^{2} / r

, banking & vertical circles are dynamics — see Laws of Motion.

More JEE Main Physics formula sheets

Frequently Asked Questions

What are the most important Kinematics formulas for JEE Main?

This Kinematics formula sheet covers all the high-yield Physics formulas, definitions and theorems you need for JEE Main, across Frame of reference, Motion in a straight line, Speed and velocity, Uniform and non-uniform motion, Average speed and instantaneous velocity — each shown with the key result and, where useful, a worked example.

Is this Kinematics formula sheet free?

Yes — the full chapter formula sheet is free to read online, no login or payment required.

How should I revise Kinematics formulas?

Blurt the Kinematics formulas from memory, then check against this sheet to find your gaps — and practise a few previous-year questions on the chapter to make sure you can apply them under time pressure.

Also useful: all formula sheetsJEE Main previous-year papersmost important chapters.