Physics
Kinematics
Kinematics Formula Sheet
All key Kinematics formulas for JEE Main & Advanced — 1D/2D motion, SUVAT equations, projectiles, relative velocity, circular motion. Fast last-minute revision.
Every must-know Kinematics formula on one page — built straight from this chapter’s topics for last-minute revision before JEE Main and Advanced.
How to use this sheet
Scan top to bottom the night before the exam. Boxed formulas are the headline results you cannot afford to forget. Sign conventions and “missing variable” tricks are where most marks are lost.
Basic Definitions (1D)
| Quantity | Symbol | Type | Formula | SI Unit |
|---|---|---|---|---|
| Distance | $d$ | Scalar | Total path length | m |
| Displacement | $\vec{s}$ | Vector | $\vec{r}_f - \vec{r}_i$ | m |
| Average speed | — | Scalar | $\dfrac{\text{Total distance}}{\text{Total time}}$ | m/s |
| Average velocity | $\vec{v}_{avg}$ | Vector | $\dfrac{\Delta \vec{s}}{\Delta t}$ | m/s |
| Instantaneous velocity | $\vec{v}$ | Vector | $\dfrac{d\vec{s}}{dt}$ | m/s |
| Acceleration | $\vec{a}$ | Vector | $\dfrac{d\vec{v}}{dt}$ | m/s² |
Distance vs displacement
Distance $\geq |\text{Displacement}|$ always. They are equal only for straight-line motion with no reversal. On a complete round trip, average velocity $= 0$ but average speed $> 0$.
Negative acceleration ≠ slowing down
What matters is the relative direction of $\vec{v}$ and $\vec{a}$:
| $\vec{v}$ and $\vec{a}$ | Motion |
|---|---|
| Same direction | Speeding up |
| Opposite directions | Slowing down |
Kinematic Equations (Constant Acceleration)
The SUVAT equations — 5 variables ($u, v, a, t, s$), 4 equations.
$$\boxed{v = u + at}$$$$\boxed{s = ut + \tfrac{1}{2}at^2}$$$$\boxed{v^2 = u^2 + 2as}$$$$\boxed{s = \frac{(u + v)}{2}\,t}$$Displacement in the nth second
$$\boxed{S_n = u + \frac{a}{2}(2n - 1)}$$Equation-selection trick
| Missing variable | Use equation |
|---|---|
| $v$ not in question | $s = ut + \tfrac{1}{2}at^2$ |
| $t$ not in question | $v^2 = u^2 + 2as$ |
| $s$ not in question | $v = u + at$ |
| $a$ not in question | $s = \tfrac{(u+v)}{2}t$ |
High-yield ratios (start from rest, u = 0)
- Distance in successive seconds: $S_1 : S_2 : S_3 : \dots = 1 : 3 : 5 : 7 : \dots$ (odd numbers)
- Total distance after each second: $s_1 : s_2 : s_3 : \dots = 1 : 4 : 9 : 16 : \dots$ (perfect squares)
When SUVAT fails
These equations only work for constant acceleration. If $a$ varies (e.g. $a = 2t$), use calculus: $v = \int a\,dt$ and $s = \int v\,dt$.
Free Fall and Vertical Motion
Free fall: $g \approx 10\ \text{m/s}^2$ (or $9.8\ \text{m/s}^2$).
Object dropped from rest
| Quantity | Formula |
|---|---|
| Velocity after time $t$ | $v = gt$ |
| Height fallen in time $t$ | $h = \tfrac{1}{2}gt^2$ |
| Velocity after falling height $h$ | $v = \sqrt{2gh}$ |
Object thrown upward (speed $u$)
| Quantity | Formula |
|---|---|
| Maximum height | $H = \dfrac{u^2}{2g}$ |
| Time to reach top | $t = \dfrac{u}{g}$ |
| Total time of flight | $T = \dfrac{2u}{g}$ |
| Speed at height $h$ | $v = \sqrt{u^2 - 2gh}$ |
Symmetry rule
Speed going up at height $h$ = speed coming down at the same height $h$. Thrown at 30 m/s → returns at 30 m/s (no air resistance).
Sign convention trap
Going up, take $a = -g$. For a ball thrown up at $u$, max height from $v^2 = u^2 + 2as$ with $a = -g$ gives $s = +\dfrac{u^2}{2g}$. Also: a ball returning to the hand has displacement = 0 but distance = 2H.
Graphical Analysis
| Graph | Slope gives | Area under curve gives |
|---|---|---|
| $x$–$t$ | Velocity $\left(v = \dfrac{dx}{dt}\right)$ | — |
| $v$–$t$ | Acceleration | Displacement |
| $a$–$t$ | — | Change in velocity |
Area sign convention
Area above the time axis = positive displacement; area below = negative. Net displacement is the algebraic sum; total distance adds magnitudes of both.
Motion in a Plane (2D)
| Quantity | Vector form | Magnitude | Direction |
|---|---|---|---|
| Position | $\vec{r} = x\hat{i} + y\hat{j}$ | $r = \sqrt{x^2 + y^2}$ | $\theta = \tan^{-1}\!\left(\tfrac{y}{x}\right)$ |
| Velocity | $\vec{v} = v_x\hat{i} + v_y\hat{j}$ | $v = \sqrt{v_x^2 + v_y^2}$ | $\theta = \tan^{-1}\!\left(\tfrac{v_y}{v_x}\right)$ |
| Acceleration | $\vec{a} = a_x\hat{i} + a_y\hat{j}$ | $a = \sqrt{a_x^2 + a_y^2}$ | — |
Independence of components
The $x$ and $y$ motions are completely independent. Apply the SUVAT equations separately to each axis:
x-axis: $v_x = u_x + a_x t$, $\;\; x = u_x t + \tfrac{1}{2}a_x t^2$, $\;\; v_x^2 = u_x^2 + 2a_x x$
y-axis: $v_y = u_y + a_y t$, $\;\; y = u_y t + \tfrac{1}{2}a_y t^2$, $\;\; v_y^2 = u_y^2 + 2a_y y$
Projectile Motion
Horizontal motion is uniform; vertical motion is uniformly accelerated ($a = g$ downward). Path is a parabola.
Horizontal projection (from height $h$, speed $u$)
Initial conditions: $u_x = u$, $u_y = 0$.
$$\boxed{T = \sqrt{\frac{2h}{g}}}$$$$\boxed{R = u\sqrt{\frac{2h}{g}} = uT}$$$$\boxed{v_{final} = \sqrt{u^2 + 2gh}}$$At time $t$: $\;x = ut$, $\;y = \tfrac{1}{2}gt^2$, $\;v_y = gt$.
Time of flight
For horizontal projection, time of flight depends only on the height $h$, not on the horizontal launch speed $u$.
Oblique projection (angle $\theta$, speed $u$, ground to ground)
Initial components: $u_x = u\cos\theta$, $u_y = u\sin\theta$.
$$\boxed{T = \frac{2u\sin\theta}{g}}$$$$\boxed{H_{max} = \frac{u^2\sin^2\theta}{2g}}$$$$\boxed{R = \frac{u^2\sin 2\theta}{g}}$$$$\boxed{y = x\tan\theta - \frac{gx^2}{2u^2\cos^2\theta}}$$| Quantity | Formula |
|---|---|
| Position at time $t$ | $x = (u\cos\theta)t$, $\;y = (u\sin\theta)t - \tfrac{1}{2}gt^2$ |
| Velocity components | $v_x = u\cos\theta$ (constant), $\;v_y = u\sin\theta - gt$ |
| Time to reach max height | $\dfrac{T}{2} = \dfrac{u\sin\theta}{g}$ |
| Trajectory (alt. form) | $y = x\tan\theta\left(1 - \dfrac{x}{R}\right)$ |
| Speed at height $h$ | $v = \sqrt{u^2 - 2gh}$ |
| Speed at time $t$ | $v = \sqrt{u^2\cos^2\theta + (u\sin\theta - gt)^2}$ |
| Velocity direction | $\tan\alpha = \dfrac{u\sin\theta - gt}{u\cos\theta}$ |
Maximum range and special cases
$$\boxed{R_{max} = \frac{u^2}{g} \quad \text{at } \theta = 45°}$$| Case | Result |
|---|---|
| Complementary angles $\theta$ and $(90° - \theta)$ | Same range $R$ |
| Projection from height $h$ (upward at angle) | $T = \dfrac{u\sin\theta + \sqrt{u^2\sin^2\theta + 2gh}}{g}$ |
| Range on plane inclined at $\alpha$ | $R = \dfrac{2u^2\sin(\theta - \alpha)\cos\theta}{g\cos^2\alpha}$ |
| Max range on incline | at $\theta = \dfrac{\pi}{4} + \dfrac{\alpha}{2}$ |
Don't confuse height-at-time-t with max height
Height at time $t$ is $y = u_y t - \tfrac{1}{2}gt^2$ — this is not $H_{max} = \dfrac{u^2\sin^2\theta}{2g}$.
Relative Velocity
Velocity of A relative to B:
$$\boxed{\vec{v}_{AB} = \vec{v}_A - \vec{v}_B}$$| Property | Relation |
|---|---|
| Reverse pair | $\vec{v}_{BA} = -\vec{v}_{AB}$ |
| Sum of pair | $\vec{v}_{AB} + \vec{v}_{BA} = 0$ |
| Chain rule | $\vec{v}_{AC} = \vec{v}_{AB} + \vec{v}_{BC}$ |
2D magnitude
$$|\vec{v}_{AB}| = \sqrt{v_A^2 + v_B^2 - 2 v_A v_B \cos\theta}$$| Angle $\theta$ between $\vec{v}_A$, $\vec{v}_B$ | $|\vec{v}_{AB}|$ | |—|—| | $0°$ (same direction) | $\lvert v_A - v_B\rvert$ | | $180°$ (opposite) | $v_A + v_B$ | | $90°$ (perpendicular) | $\sqrt{v_A^2 + v_B^2}$ |
1D quick rule
Same direction → subtract speeds. Opposite directions → add speeds.
Rain–man problem (rain vertical at $v_r$, man horizontal at $v_m$)
$$|\vec{v}_{rm}| = \sqrt{v_r^2 + v_m^2}, \qquad \tan\alpha = \frac{v_m}{v_r} \;\;(\text{angle from vertical})$$The man tilts his umbrella by $\alpha$ forward. Rain appears vertical only when he stops ($v_m = 0$); the faster he walks, the larger $\alpha$.
River crossing (width $d$, boat $v_b$ in still water, current $v_r$)
| Objective | Boat direction | Time | Drift |
|---|---|---|---|
| Shortest time | Perpendicular to bank | $\dfrac{d}{v_b}$ | $\dfrac{v_r d}{v_b}$ |
| Shortest path (no drift) | Upstream at $\theta = \sin^{-1}\!\left(\dfrac{v_r}{v_b}\right)$ | $\dfrac{d}{\sqrt{v_b^2 - v_r^2}}$ | $0$ |
Shortest-path resultant speed: $v_{res} = \sqrt{v_b^2 - v_r^2}$.
Crossing condition
A no-drift (shortest path) crossing is only possible if $v_b > v_r$ — the boat must be faster than the current.
Aircraft in wind
Ground velocity $= \vec{v}_g = \vec{v}_a + \vec{v}_w$ (airspeed plus wind velocity).
Circular Motion
Angular quantities
$$\omega = \frac{d\theta}{dt}, \qquad \alpha = \frac{d\omega}{dt}$$One revolution $= 2\pi\ \text{rad} = 360°$.
Linear–angular relations (radius $r$)
| Linear | Relation |
|---|---|
| Arc length | $s = r\theta$ |
| Linear velocity | $v = r\omega$ |
| Tangential acceleration | $a_t = r\alpha$ |
Uniform circular motion
$$\boxed{v = r\omega}$$$$\boxed{T = \frac{2\pi}{\omega} = \frac{2\pi r}{v}}$$$$\boxed{f = \frac{1}{T} = \frac{\omega}{2\pi}}$$Centripetal acceleration
$$\boxed{a_c = \frac{v^2}{r} = \omega^2 r = v\omega}$$Always directed toward the center, perpendicular to velocity.
Non-uniform circular motion
Two components — centripetal ($a_c = \dfrac{v^2}{r}$, toward center) and tangential ($a_t = \dfrac{dv}{dt} = r\alpha$, along tangent):
$$|a| = \sqrt{a_c^2 + a_t^2}, \qquad \tan\phi = \frac{a_t}{a_c}$$where $\phi$ is the angle the net acceleration makes with the radius.
Angular kinematic equations (constant $\alpha$)
| Linear form | Angular form |
|---|---|
| $v = u + at$ | $\omega = \omega_0 + \alpha t$ |
| $s = ut + \tfrac{1}{2}at^2$ | $\theta = \omega_0 t + \tfrac{1}{2}\alpha t^2$ |
| $v^2 = u^2 + 2as$ | $\omega^2 = \omega_0^2 + 2\alpha\theta$ |
Centrifugal force is not real
Centripetal force is a real force in an inertial frame (toward center). Centrifugal force is a pseudo force that only appears in the rotating frame (away from center) — it has no third-law reaction pair.
Frame of Reference
Pseudo force in a non-inertial frame accelerating at $\vec{a}_{frame}$:
$$\boxed{\vec{F}_{pseudo} = -m\,\vec{a}_{frame}}$$Apparent weight in a lift
| Lift motion | Effective weight |
|---|---|
| Accelerating up (accel. $a$) | $m(g + a)$ |
| Accelerating down (accel. $a$) | $m(g - a)$ |
One-Glance Headline Formulas
| Topic | Formula |
|---|---|
| First SUVAT | $v = u + at$ |
| Second SUVAT | $s = ut + \tfrac{1}{2}at^2$ |
| Third SUVAT | $v^2 = u^2 + 2as$ |
| nth second | $S_n = u + \tfrac{a}{2}(2n-1)$ |
| Max height (thrown up) | $H = \dfrac{u^2}{2g}$ |
| Time of flight (oblique) | $T = \dfrac{2u\sin\theta}{g}$ |
| Projectile max height | $H_{max} = \dfrac{u^2\sin^2\theta}{2g}$ |
| Projectile range | $R = \dfrac{u^2\sin 2\theta}{g}$ |
| Max range | $R_{max} = \dfrac{u^2}{g}$ at $45°$ |
| Relative velocity | $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$ |
| Linear–angular | $v = r\omega$ |
| Centripetal accel. | $a_c = \dfrac{v^2}{r} = \omega^2 r$ |
Related Topics
- Kinematic Equations — full derivations and worked patterns
- Projectile Motion — horizontal, oblique, and inclined-plane cases
- Relative Velocity — rain–man and river-crossing problems
- Circular Motion — angular kinematics and centripetal acceleration
- Graphical Analysis — slope and area interpretation