Kinematics — Complete Formula

⭐ – Most used in JEE
⚠️ – Common Mistake
💡 – Memory Hint

Basic Definitions

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Speed	[\text{Speed} = \dfrac{\text{Distance}}{\text{Time}}]	—	m/s	Scalar quantity (no direction)
Velocity	[v = \dfrac{s}{t}]	v = velocity, s = displacement, t = time	m/s	Use displacement, NOT distance ⚠️
Acceleration	[a = \dfrac{v – u}{t}]	u = initial velocity, v = final velocity	m/s²	Valid only for uniform acceleration
Average Speed	[\text{Avg Speed} = \dfrac{\text{Total Distance}}{\text{Total Time}}]	—	m/s	Always ≥ average velocity ⚠️
Average Velocity	[v_{avg} = \dfrac{\text{Total Displacement}}{\text{Total Time}}]	—	m/s	Can be zero even if motion occurs ⚠️
Instantaneous Velocity	[v = \dfrac{ds}{dt}]	—	m/s	Velocity at a specific instant ⭐

Equations of Motion (Uniform Acceleration)

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
First Equation	[v = u + at]	u = initial velocity, v = final velocity	m/s	Most basic equation ⭐
Second Equation	[s = ut + \dfrac{1}{2}at^2]	s = displacement	m	Use when time is given
Third Equation	[v^2 = u^2 + 2as]	—	m²/s²	Use when time is NOT given ⭐
Displacement (nth second)	[s_n = u + \dfrac{a}{2}(2n – 1)]	n = second number	m	Useful in discrete motion questions
Average Velocity (uniform acc.)	[v_{avg} = \dfrac{u + v}{2}]	—	m/s	Only for uniform acceleration ⚠️

💡 Memory Hint:
If time is missing → use [v^2 = u^2 + 2as]
If time is given → use [s = ut + \dfrac{1}{2}at^2]

Graph-Based Relations

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Velocity from Position Graph	[v = \dfrac{ds}{dt}]	slope of s–t graph	m/s	Slope gives velocity ⭐
Acceleration from Velocity Graph	[a = \dfrac{dv}{dt}]	slope of v–t graph	m/s²	Slope gives acceleration ⭐
Displacement from Velocity Graph	[s = \int v , dt]	area under v–t graph	m	Area = displacement ⭐
Acceleration from Position Graph	[a = \dfrac{d^2 s}{dt^2}]	—	m/s²	Second derivative

💡 Memory Hint:

Slope → derivative → velocity/acceleration
Area → integration → displacement

Free Fall (Under Gravity)

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Acceleration due to Gravity	[g \approx 9.8]	—	m/s²	Always downward ⭐
Velocity in Free Fall	[v = u + gt]	—	m/s	Replace ‘a’ with ‘g’
Displacement in Free Fall	[s = ut + \dfrac{1}{2}gt^2]	—	m	Same motion equations
Velocity-Height Relation	[v^2 = u^2 + 2gs]	—	m²/s²	Use when time not given
Body Dropped from Rest	[v = gt,\quad s = \dfrac{1}{2}gt^2]	u = 0	—	Most common case ⭐

💡 Memory Hint:
Free fall = just motion equations with [a → g]

Relative Motion (1D Basics)

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Relative Velocity	[v_{AB} = v_A – v_B]	velocity of A w.r.t B	m/s	Think “A minus B” ⭐
Same Direction	[v_{rel} = v_A – v_B]	—	m/s	Sign matters ⚠️
Opposite Direction	[v_{rel} = v_A + v_B]	—	m/s	Speeds add up ⭐

💡 Memory Hint:

Same direction → subtract
Opposite direction → add

Perfect—now we complete Kinematics with the remaining high-weight sections in the same locked Ucale format.

Projectile Motion (2D Kinematics)

Basic Components of Motion

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Horizontal Velocity	[v_x = u\cos\theta]	u = initial velocity, θ = angle of projection	m/s	Remains constant (no acceleration) ⭐
Vertical Velocity	[v_y = u\sin\theta – gt]	—	m/s	Changes due to gravity
Resultant Velocity	[v = \sqrt{v_x^2 + v_y^2}]	—	m/s	Vector combination

💡 Memory Hint:
Break motion into independent x and y components

Time of Flight, Maximum Height, Range

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Time of Flight	[T = \dfrac{2u\sin\theta}{g}]	—	s	Depends only on vertical motion ⭐
Maximum Height	[H = \dfrac{u^2 \sin^2\theta}{2g}]	—	m	At top: vertical velocity = 0 ⭐
Horizontal Range	[R = \dfrac{u^2 \sin 2\theta}{g}]	—	m	Maximum at θ = 45° ⭐
Time to Reach Top	[t = \dfrac{u\sin\theta}{g}]	—	s	Half of total time ⚠️

💡 Memory Hint:

Range → sin2θ (symmetry concept)
Max height → depends on (sinθ)²

Trajectory Equation

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Trajectory Path	[y = x\tan\theta – \dfrac{g x^2}{2u^2 \cos^2\theta}]	—	m	Always a parabola ⭐

💡 Memory Hint:
Projectile path = parabolic curve

Special Cases

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Horizontal Projection	[T = \sqrt{\dfrac{2h}{g}}]	h = height	s	Initial vertical velocity = 0 ⭐
Range from Height	[R = u\cos\theta \cdot T]	—	m	Use time first ⚠️
Same Range Angles	[\theta \text{ and } (90^\circ – \theta)]	—	—	Give same range ⭐

💡 Memory Hint:
Complementary angles → same range

Relative Motion (2D Advanced — River, Rain, etc.)

Vector Form (Most Important)

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Relative Velocity	[\vec{v}_{AB} = \vec{v}_A – \vec{v}_B]	velocity of A w.r.t B	m/s	Always treat as vectors ⭐
Magnitude	[v_{rel} = \sqrt{v_x^2 + v_y^2}]	—	m/s	Use Pythagoras for perpendicular motion

💡 Memory Hint:
Relative motion = vector subtraction

River Boat Problems

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Resultant Velocity	[v = \sqrt{v_b^2 + v_r^2}]	v_b = boat velocity, v_r = river velocity	m/s	When directions are perpendicular ⭐
Time to Cross River	[t = \dfrac{d}{v_b}]	d = width	s	Depends only on perpendicular velocity ⚠️
Drift (Downstream Shift)	[x = v_r \cdot t]	—	m	Due to river flow
Shortest Path Condition	Boat aimed upstream	—	—	Cancel river drift ⭐
Minimum Time Condition	Boat perpendicular to river	—	—	Fastest crossing ⚠️

💡 Memory Hint:

Minimum time ≠ shortest path (very common trap ⚠️)

Rain Problems

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Apparent Velocity of Rain	[\vec{v}_{rain/man}] [= \vec{v}_{rain} – \vec{v}_{man}]	—	m/s	Relative velocity concept ⭐
Angle of Rain	[\tan\theta = \dfrac{v_{horizontal}}{v_{vertical}}]	—	—	Direction seen by observer
Condition to Avoid Rain	Move with rain direction	—	—	Relative velocity becomes vertical ⚠️

💡 Memory Hint:
Rain problems = what observer sees → relative velocity

Aeroplane / Wind Problems

Concept	Formula	Symbols Meaning	SI Units	Key Notes / Tricks
Resultant Velocity	[\vec{v} = \vec{v}_{plane} + \vec{v}_{wind}]	—	m/s	Vector addition ⭐
Drift Angle	Depends on wind direction	—	—	Plane deviates due to wind ⚠️
Straight Path Condition	Adjust angle against wind	—	—	Cancel drift ⭐

💡 Memory Hint:
Wind problems = same as river flow logic

Unlock the full course today

Get full access to all videos and content.

Explore