Kinematics formulas for JEE Main & Advanced

19 formula entries for Kinematics, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Distance vs Displacement

Displacement

\vec{s} = {\vec{r}}_{f} - {\vec{r}}_{i}

When to use. Always distinguish; displacement is key for velocity, distance for speed.

Trap. Using distance in place of displacement in velocity calculations.

Speed vs Velocity

Average speed = \frac{total distance}{Δ t}

{\vec{v}}_{a v g} = \frac{Δ \vec{s}}{Δ t}

When to use. Use speed for path length, velocity for position change.

Trap. Assuming average speed equals magnitude of average velocity.

Instantaneous Velocity & Acceleration

\vec{v} = \frac{d \vec{r}}{d t}

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

When to use. When motion is described by a function of time.

Trap. Forgetting that acceleration can be non-zero even when speed is constant (e.g., circular motion).

Equations of Motion (Constant Acceleration)

v = u + a t

;

s = u t + \frac{1}{2} a t^{2}

;

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

When to use. Any problem with uniform acceleration (e.g., free fall, motion on incline).

Trap. Using these for non-constant acceleration (requires calculus).

Graphical Analysis: v-t and x-t

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

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

Δ x = \int v d t

When to use. When given graphs or asked to interpret motion from graphs.

Trap. Confusing area under v-t with distance (area gives displacement; distance is absolute area).

Relative Motion in 1D

{\vec{v}}_{A B} = {\vec{v}}_{A} - {\vec{v}}_{B}

When to use. When two objects move along the same line (e.g., overtaking, collision).

Trap. Sign errors: subtract velocities in correct order.

Projectile Motion Basics

x = u_{x} t

y = u_{y} t - \frac{1}{2} g t^{2}

When to use. Any object launched near Earth's surface with only gravity.

Trap. Assuming horizontal acceleration is non-zero.

Projectile: Time of Flight, Range, Max Height

T = \frac{2 u \sin θ}{g}

;

R = \frac{u^{2} \sin 2 θ}{g}

;

H = \frac{u^{2} \sin^{2} θ}{2 g}

When to use. For level ground, no air resistance.

Trap. Complementary angles ( $θ$ and $90^{\circ} - θ$ ) give the same range since $\sin 2 θ = \sin (180^{\circ} - 2 θ)$ ; maximum range is at $45^{\circ}$ .

Projectile on Inclined Plane

For incline angle

α

, launch angle

θ

relative to incline:

a_{x} = - g \sin α

a_{y} = - g \cos α

When to use. When projectile is launched up or down an incline.

Trap. Using standard projectile formulas directly without adjusting for incline.

Uniform Circular Motion

a_{c} = \frac{v^{2}}{r} = ω^{2} r

v = ω r

T = \frac{2 π}{ω}

When to use. Any object moving in a circular path at constant speed.

Trap. Thinking acceleration is zero because speed is constant.

Non-Uniform Circular Motion

a_{t} = \frac{d v}{d t}

a_{c} = \frac{v^{2}}{r}

a_{net} = \sqrt{a_{t}^{2} + a_{c}^{2}}

When to use. When speed is not constant (e.g., vertical loop).

Trap. Forgetting tangential component when speed changes.

Relative Motion in 2D

{\vec{v}}_{A B} = {\vec{v}}_{A} - {\vec{v}}_{B}

When to use. When two objects move in a plane (e.g., rain-man, boat-river).

Trap. Adding instead of subtracting vectors.

River-Boat Problems

{\vec{v}}_{b g} = {\vec{v}}_{b w} + {\vec{v}}_{w g}

; Time to cross:

t = \frac{d}{v_{b w} \cos θ}

; Drift:

x = (v_{w g} - v_{b w} \sin θ) t

When to use. Boat crossing a river with current.

Trap. Confusing angle of boat with direction of resultant velocity.

Rain-Man Problems

{\vec{v}}_{r m} = {\vec{v}}_{r} - {\vec{v}}_{m}

When to use. When a person moves in rain and needs umbrella direction.

Trap. Using ${\vec{v}}_{r} + {\vec{v}}_{m}$ instead of subtraction.

Calculus in Kinematics

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

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

Δ x = \int_{t_{1}}^{t_{2}} v d t

Δ v = \int_{t_{1}}^{t_{2}} a d t

When to use. When acceleration is a function of time, velocity, or position.

Trap. Forgetting integration constants (initial conditions).

Motion Graphs: x-t, v-t, a-t Interconversion

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

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

Δ x = \int v d t

Δ v = \int a d t

When to use. When given one graph and asked to sketch another or find quantities.

Trap. Confusing area under v-t with distance (area gives displacement; distance is absolute area).

Free Fall Under Gravity

v = u \pm g t

s = u t \pm \frac{1}{2} g t^{2}

v^{2} = u^{2} \pm 2 g s

When to use. Any object dropped or thrown vertically near Earth.

Trap. Using $g$ positive without sign convention.

Stopping Distance and Reaction Time

s_{total} = v_{0} t_{r} + \frac{v_{0}^{2}}{2 a}

(where

a

is deceleration)

When to use. Problems involving vehicle braking or safety.

Trap. Forgetting the reaction time component.

Average vs Instantaneous Quantities

v_{a v g} = \frac{Δ x}{Δ t}

v = \lim_{Δ t \to 0} \frac{Δ x}{Δ t}

When to use. When time interval is given vs specific time.

Trap. Using average formulas for instantaneous values.

Values to remember

$g = 9.8 {m/s}^{2}$ (or $10 {m/s}^{2}$ in some problems) Equations of Motion (Constant Acceleration)
$g = 9.8 {m/s}^{2}$ Projectile Motion Basics

Read the full chapter: Kinematics for JEE, with what JEE asks and the common traps.