Physics · Mechanics and Waves

Kinematics formulas for JEE

Every Kinematics formula you need for JEE, grouped by concept.

43 formulas6 concepts

01Motion in a Straight Line 02Uniformly Accelerated Motion 03Calculus in Kinematics 04Motion in a Plane 05Projectile Motion 06Uniform Circular Motion

Motion in a Straight Line

4 formulas

Average Speed

v_{avg} = \frac{\text{Total Path Length}}{\Delta t}

Total path length covered divided by total time interval.

applies whenValid for any type of 1D motion.

kinematicsspeed

Average Speed (Equal Distances)

v_{avg} = \frac{2v_1 v_2}{v_1 + v_2}

Average speed when an object covers two equal distances at different speeds (Harmonic Mean).

applies whenPath split into two equal distance halves.

kinematicsaverage_speedjee-advanced

Average Speed (Equal Times)

v_{avg} = \frac{v_1 + v_2}{2}

Average speed when an object travels for two equal time intervals at different speeds (Arithmetic Mean).

applies whenTime interval split into two equal halves.

kinematicsaverage_speedjee-advanced

Average Velocity Vector

\mathbf{v}_{avg} = \frac{\Delta\mathbf{r}}{\Delta t}

The ratio of total displacement to the corresponding time interval.

kinematicsvelocityaverage

Uniformly Accelerated Motion

13 formulas

Instantaneous Acceleration Vector

\mathbf{a} = \lim_{\Delta t \to 0} \frac{\Delta\mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}

The limiting value of average acceleration as the time interval approaches zero.

kinematicsaccelerationcalculus

Average Acceleration Vector

\mathbf{a}_{avg} = \frac{\Delta\mathbf{v}}{\Delta t}

The change in velocity divided by the time interval.

kinematicsaccelerationaverage

First Kinematic Equation (Vector)

\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t

Velocity of an object at time t under constant acceleration.

applies whenAcceleration vector must be strictly constant in magnitude and direction.

kinematicsconstant_acceleration

Second Kinematic Equation (Vector)

\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2}\mathbf{a}t^2

Position vector of an object at time t under constant acceleration.

applies whenAcceleration vector must be strictly constant.

kinematicsconstant_acceleration

Third Kinematic Equation

v^2 = v_0^2 + 2a(x - x_0)

Velocity-position relation for uniformly accelerated motion.

applies whenAcceleration 'a' must be constant.

kinematicsconstant_acceleration

Displacement using Average Velocity

x - x_0 = \left( \frac{v + v_0}{2} \right) t

Displacement defined by the arithmetic mean of initial and final velocities.

applies whenAcceleration 'a' must be constant.

kinematicsconstant_acceleration

Free Fall Displacement

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

Displacement of an object dropped from rest under gravity.

applies whenObject released from rest; upward direction is positive; negligible air resistance.

kinematicsfree_fallworked_example

Free Fall Velocity-Displacement

v^2 = -2gy

Velocity squared of an object dropped from rest in terms of its displacement.

applies whenObject released from rest; upward direction is positive; negligible air resistance.

kinematicsfree_fallworked_example

Free Fall Velocity

v = -gt

Velocity of an object dropped from rest under gravity.

applies whenObject released from rest; upward direction is positive; negligible air resistance.

kinematicsfree_fallworked_example

Galileo's Law of Odd Numbers

y_1 : y_2 : y_3 \dots = 1 : 3 : 5 \dots

Ratio of distances traversed during equal, successive time intervals by a body falling from rest.

applies whenObject falling from rest under constant acceleration.

kinematicsfree_fallworked_example

Displacement in the n-th Second

S_n = u + \frac{a}{2}(2n - 1)

Distance covered by an accelerating object during the n-th second of its motion.

applies whenConstant acceleration.

kinematicsjee-advanced

Reaction Time

t_r = \sqrt{\frac{2d}{g}}

Time taken to respond, modeled via the distance a dropped ruler falls.

applies whenFree fall from rest.

kinematicsreaction_timeworked_example

Stopping Distance

d_s = \frac{-v_0^2}{2a}

Distance a vehicle travels before stopping when brakes are applied.

applies whenConstant deceleration 'a' (where 'a' is negative).

kinematicsstopping_distanceworked_example

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Calculus in Kinematics

3 formulas

Instantaneous Acceleration Vector

\mathbf{a} = \lim_{\Delta t \to 0} \frac{\Delta\mathbf{v}}{\Delta t} = \frac{d\mathbf{v}}{dt} = \frac{d^2\mathbf{r}}{dt^2}

The limiting value of average acceleration as the time interval approaches zero.

kinematicsaccelerationcalculus

Acceleration (Position Derivative)

a = v \frac{dv}{dx}

Acceleration expressed using the chain rule with respect to position.

applies whenUseful when velocity is given as a function of position.

kinematicsaccelerationcalculusworked_example

Instantaneous Velocity Vector

\mathbf{v} = \lim_{\Delta t \to 0} \frac{\Delta\mathbf{r}}{\Delta t} = \frac{d\mathbf{r}}{dt}

The limiting value of average velocity as the time interval approaches zero.

kinematicsvelocitycalculus

Motion in a Plane

9 formulas

Average Acceleration Vector

\mathbf{a}_{avg} = \frac{\Delta\mathbf{v}}{\Delta t}

The change in velocity divided by the time interval.

kinematicsaccelerationaverage

Average Velocity Vector

\mathbf{v}_{avg} = \frac{\Delta\mathbf{r}}{\Delta t}

The ratio of total displacement to the corresponding time interval.

kinematicsvelocityaverage

First Kinematic Equation (Vector)

\mathbf{v} = \mathbf{v}_0 + \mathbf{a}t

Velocity of an object at time t under constant acceleration.

applies whenAcceleration vector must be strictly constant in magnitude and direction.

kinematicsconstant_acceleration

Second Kinematic Equation (Vector)

\mathbf{r} = \mathbf{r}_0 + \mathbf{v}_0 t + \frac{1}{2}\mathbf{a}t^2

Position vector of an object at time t under constant acceleration.

applies whenAcceleration vector must be strictly constant.

kinematicsconstant_acceleration

Relative Velocity in 2D

\mathbf{v}_{AB} = \mathbf{v}_A - \mathbf{v}_B

Velocity of object A as observed from the reference frame of object B.

applies whenVelocities must be measured from a common reference frame.

kinematicsrelative_motion

Law of Cosines for Vector Addition

R = \sqrt{A^2 + B^2 + 2AB \cos\theta}

Magnitude of the resultant vector when adding two vectors separated by an angle theta.

applies whenVectors A and B must be joined tail-to-tail with angle theta between them.

vectorsadditionresultant

Direction of Resultant Vector

\tan\alpha = \frac{B \sin\theta}{A + B \cos\theta}

Calculates the angle alpha that the resultant vector R makes with the vector A.

vectorsdirectionresultant

Magnitude of a Vector (3D)

A = \sqrt{A_x^2 + A_y^2 + A_z^2}

Calculates the absolute length or magnitude of a vector from its rectangular components.

applies whenApplicable for Cartesian coordinate system.

vectorsmagnitudecomponents

Law of Sines for Vectors

\frac{R}{\sin\theta} = \frac{A}{\sin\beta} = \frac{B}{\sin\alpha}

Relates the magnitudes of vectors to the sines of their opposite angles in a vector addition triangle.

vectorstriangle_law

Projectile Motion

8 formulas

Range on an Incline

R = \frac{u^2}{g \cos^2\beta} [\sin(2\alpha-\beta) - \sin\beta]

Distance traveled along an inclined plane.

applies whenIncline angle beta, projection angle alpha with horizontal, projected upwards along incline.

projectileinclinejee-advanced

Time of Flight on an Incline

T = \frac{2u \sin(\alpha-\beta)}{g \cos\beta}

Time of flight for a projectile launched up an inclined plane.

applies whenIncline angle beta, projection angle alpha with horizontal.

projectileinclinejee-advanced

Projectile Maximum Height

h_m = \frac{v_0^2 \sin^2\theta_0}{2g}

The maximum vertical height attained by a projectile.

applies whenLaunched from a horizontal plane, constant gravity, no air resistance.

projectileheight

Maximum Horizontal Range

R_{m} = \frac{v_0^2}{g}

Maximum range possible for a given velocity, achieved when projected at 45 degrees.

applies whenLanding elevation equals launch elevation.

projectilerange

Projectile Horizontal Range

R = \frac{v_0^2 \sin(2\theta_0)}{g}

Total horizontal distance covered by a projectile over ground.

applies whenLanding elevation equals launch elevation, constant gravity.

projectilerange

Projectile Time of Flight

T_f = \frac{2v_0 \sin\theta_0}{g}

Total time duration the projectile remains in the air before landing back on the initial vertical level.

applies whenLanding elevation equals launch elevation, constant gravity.

projectiletime

Equation of Projectile Trajectory

y = x \tan\theta_0 - \frac{g x^2}{2(v_0 \cos\theta_0)^2}

Parabolic path equation relating vertical position (y) to horizontal position (x).

applies whenConstant vertical gravity, no air resistance.

projectiletrajectory

Trajectory Equation via Horizontal Range

y = x \tan\theta \left(1 - \frac{x}{R}\right)

Alternative form of the projectile trajectory utilizing the horizontal range.

applies whenConstant gravity, no air resistance.

projectiletrajectoryjee-advanced

Uniform Circular Motion

6 formulas

Angular Speed

\omega = \frac{\Delta\theta}{\Delta t}

Time rate of change of angular displacement.

circular_motionangular_velocity

Centripetal Acceleration (Angular)

a_c = \omega^2 R = 4\pi^2 \nu^2 R

Centripetal acceleration expressed in terms of angular velocity and frequency.

applies whenCircular motion.

circular_motionacceleration

Centripetal Acceleration

a_c = \frac{v^2}{R}

Acceleration directed towards the center in a circular path.

applies whenUniform or instantaneous non-uniform circular motion.

circular_motionacceleration

Linear and Angular Velocity Relation

v = \omega R

Relation between linear speed v and angular speed omega in circular motion.

applies whenCircular motion.

circular_motionvelocity

Total Acceleration in Circular Motion

a = \sqrt{a_c^2 + a_t^2} = \sqrt{\left(\frac{v^2}{R}\right)^2 + \left(\frac{dv}{dt}\right)^2}

Net magnitude of acceleration when speed is changing on a circular path.

applies whenNon-uniform circular motion.

circular_motionaccelerationjee-advanced

Radius of Curvature

R_c = \frac{v^3}{|\mathbf{v} \times \mathbf{a}|}

Radius of the best-fitting circle to a curve at a given point.

applies whenGeneral 2D curvilinear motion.

kinematicscurvaturejee-advanced

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