- ⭐ – 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
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