Physics · Mechanics and Waves

Laws of Motion formulas for JEE

Every Laws of Motion formula you need for JEE, grouped by concept.

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QuestionState the formula — Newton's Second Law (Constant Mass)

Reveal answer

Newton's Second Law (Constant Mass)

\mathbf{F} = m\mathbf{a}

Net force is proportional to the product of mass and acceleration.applies whenConstant mass system in an inertial frame.dynamicsforceacceleration

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All 30 Laws of Motion formulas

Newton's Laws of Motion

11 formulas

Newton's Second Law (Constant Mass)

\mathbf{F} = m\mathbf{a}

Net force is proportional to the product of mass and acceleration.

applies whenConstant mass system in an inertial frame.

dynamicsforceacceleration

Newton's Second Law (General Form)

\mathbf{F} = \frac{d\mathbf{p}}{dt}

Force equals the rate of change of momentum over time.

applies whenInertial frame of reference.

dynamicsforcecalculus

Impulse Momentum Theorem

\mathbf{J} = \mathbf{F}_{avg} \Delta t = \Delta\mathbf{p}

Impulse is the product of a large force acting over a short time interval, equating to the change in momentum.

applies whenForce acts for a short duration where position change is negligible.

dynamicsimpulsemomentum

Conservation of Linear Momentum

\mathbf{p}_{A} + \mathbf{p}_{B} = \mathbf{p}'_{A} + \mathbf{p}'_{B}

Total initial momentum equals total final momentum for an isolated system.

applies whenNet external force acting on the system is zero.

dynamicsmomentumconservation

Momentum

\mathbf{p} = m\mathbf{v}

Definition of linear momentum as the product of mass and velocity.

applies whenNon-relativistic speeds.

dynamicsmomentumvectors

Newton's Third Law

\mathbf{F}_{AB} = -\mathbf{F}_{BA}

To every action, there is always an equal and opposite reaction.

applies whenForces act on different bodies simultaneously.

dynamicslaws_of_motionvectors

Pseudo Force (D'Alembert's Principle)

\mathbf{F}_{pseudo} = -m\mathbf{a}_0

Fictitious force acting on a mass observed from a non-inertial (accelerating) reference frame.

applies whenUsed strictly when analyzing dynamics from an accelerated reference frame.

dynamicspseudo_forcejee-advanced

Thrust on a Rocket

F_{thrust} = v_{rel} \frac{dm}{dt}

Upward thrust force experienced by a rocket due to the continuous ejection of exhaust gases.

applies whenRelative exhaust velocity is constant.

dynamicsvariable_massjee-advanced

Rocket Velocity (With Gravity)

v = v_0 + v_{rel} \ln\left(\frac{m_0}{m}\right) - gt

Velocity of a rocket at any instant, taking a uniform downward gravitational field into account.

applies whenUniform gravitational field; constant upward relative exhaust velocity.

dynamicsvariable_massjee-advanced

Rocket Velocity (No Gravity)

v = v_0 + v_{rel} \ln\left(\frac{m_0}{m}\right)

Velocity of a rocket at any given instant, ignoring external forces like gravity and air resistance.

applies whenZero gravity and negligible air drag.

dynamicsvariable_massjee-advanced

Variable Mass System Equation

\mathbf{F}_{ext} + \mathbf{v}_{rel}\frac{dm}{dt} = m\frac{d\mathbf{v}}{dt}

Newton's second law applied to a system losing or gaining mass (e.g., a rocket).

applies whenSystem where mass changes over time;

v_{rel}

is relative velocity of escaping mass.

dynamicsvariable_massjee-advanced

Applications of Newton's Laws

7 formulas

Atwood Machine Acceleration

a = \frac{m_1 - m_2}{m_1 + m_2}g

Acceleration of two masses connected by a string over an ideal pulley.

applies whenMassless inextensible string, frictionless and massless pulley,

m_1 > m_2

dynamicspulleysacceleration

Atwood Machine Tension

T = \frac{2m_1 m_2}{m_1 + m_2}g

Tension in the string connecting two masses over an ideal pulley.

applies whenMassless inextensible string, frictionless and massless pulley.

dynamicspulleystension

Contact Force Between 2 Blocks

f_{contact} = \frac{m_2 F}{m_1 + m_2}

Normal contact force between two adjacent masses m1 and m2 when a horizontal force F pushes on m1.

applies whenBlocks are pushed together on a frictionless horizontal surface.

dynamicscontact_forceconnected_bodiesjee-advanced

Equilibrium of Concurrent Forces

\sum \mathbf{F} = \mathbf{F}_1 + \mathbf{F}_2 + \mathbf{F}_3 = 0

The vector sum of all concurrent forces acting on a particle at rest or moving uniformly must be zero.

applies whenNet external force is zero.

staticsequilibriumvectors

Spring Restoring Force

\mathbf{F} = -k\mathbf{x}

The restoring force is proportional to the displacement from the unstretched state.

applies whenIdeal spring, valid only for small displacements.

dynamicsspringshookes_law

Tension in Connected Blocks

T = \frac{(m_2 + m_3)F}{m_1 + m_2 + m_3}

Tension in the string pulling trailing masses (m2 + m3) when force F pulls a system of m1, m2, m3.

applies whenMassless strings, frictionless surface; force applied to the leading mass m1.

dynamicstensionconnected_bodiesjee-advanced

Impulse from Oblique Wall Bounce

J = 2mu \cos \theta

Impulse delivered by a wall when a ball collides elastically at an angle theta to the normal.

applies whenElastic collision; theta is the angle with the normal.

dynamicsimpulsecollisions

Friction

6 formulas

Angle of Repose

\theta_{max} = \tan^{-1}(\mu_s)

The maximum angle of an inclined plane at which a block remains stationary.

applies whenObject on the verge of sliding down the incline due to gravity.

frictionstaticsincline

Sliding Friction

f_k = \mu_k N

Kinetic friction opposing actual relative motion between surfaces.

applies whenRelative motion exists between the two surfaces in contact.

frictionkinetics

Limiting Static Friction

(f_s)_{max} = \mu_s N

The maximum possible force of static friction before sliding begins.

applies whenBody is strictly on the verge of slipping.

frictionstaticslimiting

Static Friction Range

f_s \le \mu_s N

Static friction is a self-adjusting force up to a maximum limit.

applies whenBody is at rest or impending motion relative to the surface.

frictionstatics

Acceleration Sliding Down Incline

a = g(\sin\theta - \mu_k \cos\theta)

Net acceleration of a block sliding down a rough inclined plane.

applies whenBlock is actively moving down the incline.

dynamicsfrictioninclinejee-advanced

Retardation Sliding Up Incline

a = g(\sin\theta + \mu_k \cos\theta)

Net deceleration of a block projected up a rough inclined plane.

applies whenBlock is actively moving up the incline.

dynamicsfrictioninclinejee-advanced

Dynamics of Circular Motion

6 formulas

Banked road speed

v_0 = \sqrt{Rg\tan\theta}

Optimum speed for no friction dependency on a banked curve.

applies whenIdeal speed to eliminate lateral friction and tyre wear.

dynamicsbankedcircular_motion

Maximum Speed on Banked Road

v_{max} = \sqrt{R g \left(\frac{\mu_s + \tan \theta}{1 - \mu_s \tan \theta}\right)}

Maximum permissible speed on a banked curve considering both banking angle and static friction.

applies whenVehicle at the verge of slipping outward up the incline.

dynamicsbankedfriction

Centripetal Force

f_c = \frac{mv^2}{R} = m\omega^2 R

The net inward radial force required to keep an object moving in a circle.

applies whenUniform or non-uniform circular motion (radial component).

dynamicscircular_motion

Conical Pendulum Angular Velocity

\omega = \sqrt{\frac{g}{L\cos\theta}}

Angular velocity of a mass moving in a horizontal circle suspended by a string.

applies whenUniform horizontal circular motion; L is string length, theta is angle with vertical.

dynamicscircular_motionjee-advanced

Level road max speed

v_{max} = \sqrt{\mu_s R g}

Maximum safe speed for a vehicle taking a circular turn on a flat, unbanked road.

applies whenCentripetal force is provided purely by static friction.

dynamicscircular_motionfriction

Vertical circle condition

v_{bottom} = \sqrt{5gR}

Minimum velocity required at the lowest point to complete a vertical circle attached to a string.

applies whenMass attached to a flexible massless string; tension

\ge 0

at the top.

dynamicscircular_motionjee-advanced

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