Formula Bank & Dimensional Analysis

Every formula you need. Searchable. With dimensions. With conditions. With copy button. This is your exam-day cheat sheet — but built for understanding, not memorization.

30+ Formulas Dimensions Verified Kinematics Calculator

⚡ Equations of Motion (Uniform Acceleration)

Valid ONLY when acceleration is constant (a = const). For variable acceleration, use calculus.

#	Formula	Variables	Dimension	When to Use
1	v = u + at	v, u, a, t (no s)	[LT⁻¹]	t is known, s not asked
2	s = ut + ½at²	s, u, a, t (no v)	[L]	v not known, t is known
3	v² = u² + 2as	v, u, a, s (no t)	[L²T⁻²]	t is absent (most common)
4	s = (u + v)t / 2	s, u, v, t (no a)	[L]	a not given, u & v known
5	sₙ = u + a(n − ½)	sₙ, u, a, n	[L]	Distance in nth second only

📐 Velocity & Acceleration Definitions

#	Formula	Variables	Dimension	When to Use
6	v_avg = Δx / Δt	v_avg, x, t	[LT⁻¹]	Always valid, any motion
7	v = dx/dt	v, x, t	[LT⁻¹]	Instantaneous velocity (calculus)
8	a = dv/dt = d²x/dt²	a, v, x, t	[LT⁻²]	Instantaneous acceleration
9	a = v · dv/dx	a, v, x	[LT⁻²]	When velocity is function of position
10	Speed_avg = Total Distance / Total Time	Speed, distance, time	[LT⁻¹]	Different from velocity_avg

🌍 Free Fall & Vertical Motion (g = 10 m/s²)

#	Formula	Condition	Notes
11	v = u − gt	Up = +ve convention	Object thrown up; g negative
12	H = u² / 2g	Max height reached	At highest point: v = 0
13	t_top = u / g	Time to reach max height	When v = 0
14	T = 2u / g	Total time of flight	Return to same level
15	v_impact = √(u² + 2gH)	Object thrown from height H	H is height of building/cliff
16	h = ½gt² (dropped from rest)	u = 0, free fall	Down = +ve, most common case

🔄 Relative Motion (1D)

#	Formula	Meaning	Application
17	v_AB = v_A − v_B	Velocity of A w.r.t B	Train/boat/swimmer problems
18	a_AB = a_A − a_B	Acceleration of A w.r.t B	Both objects accelerating
19	t_meet = d / (v_A + v_B)	Time to meet (toward each other)	Train collision / meeting problems
20	t_overtake = d / (v_A − v_B)	Time to overtake (same direction)	v_A > v_B, A chasing B

📊 Graph Interpretation Formulas

#	Graph	Slope Gives	Area Gives	Key Insight
21	x–t graph	Velocity (v)	N/A	Horizontal = rest; Curve = acceleration
22	v–t graph	Acceleration (a)	Displacement (s)	Area above x-axis = +s; below = −s
23	a–t graph	Not meaningful	Change in velocity (Δv)	Area = Δv (impulse in force context)
24	v²–x graph	2 × acceleration	—	slope of v² vs x = 2a (from v²=u²+2as)

🔭 Dimensional Analysis Guide

[L] = Metre (m)

Length, displacement, distance

[T] = Second (s)

Time

[M] = Kilogram (kg)

Mass (not in 1D motion)

Dimensions of Key Quantities

Quantity	Symbol	SI Unit	Dimension
Position/Displacement	x, s	metre (m)	[L]
Velocity	v, u	m/s	[LT⁻¹]
Acceleration	a, g	m/s²	[LT⁻²]
Time	t	second (s)	[T]
Speed	\|v\|	m/s	[LT⁻¹]
Jerk (da/dt)	j	m/s³	[LT⁻³]

🔬 Dimensional Analysis Trick

To verify any kinematic formula: both sides must have the same dimension.
Check: s = ut + ½at² → [L] = [LT⁻¹·T] + [LT⁻²·T²] = [L] + [L] ✓
If dimensions don't match → formula is wrong. Use this to eliminate wrong options in MCQs instantly.

🧮 Kinematics Calculator

Initial Velocity u (m/s)

Acceleration a (m/s²)

Time t (s)

Displacement s (m) [for v²]

Find:

📐 Active Formulas

v = u + at

s = ut + ½at²

v² = u² + 2as

v_avg = (u + v) / 2

🎯 Calculator Tip

Always check: is acceleration constant? If not, these formulas don't apply. Use integration instead.

⭐ Special & Advanced Formulas

#	Formula	Topic	Level
25	x = ∫v dt	Position from velocity	JEE Adv
26	v = ∫a dt	Velocity from acceleration	JEE Adv
27	v dv/dx = a(x)	When a is function of x	JEE Adv
28	v² = u² + 2∫a dx	Non-uniform acceleration	JEE Adv
29	1 : 3 : 5 : 7 ... ratio	Distances in successive seconds (u=0)	NEET/JEE
30	1 : 4 : 9 : 16 ratio	Total distances after 1s, 2s, 3s... (u=0)	NEET/JEE

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