Physics · Mechanics and Waves

Laws of Motion formulas for JEE

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

FormulasRevision notes
30 formulas4 concepts
01Newton's Laws of Motion02Applications of Newton's Laws03Friction04Dynamics of Circular Motion
01

Newton's Laws of Motion

11 formulas

Newton's Second Law (Constant Mass)

F=ma\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)

F=dpdt\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

J=FavgΔt=Δp\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

pA+pB=pA′+pB′\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

p=mv\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

FAB=−FBA\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)

Fpseudo=−ma0\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

Fthrust=vreldmdtF_{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=v0+vrelln⁡(m0m)−gtv = 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=v0+vrelln⁡(m0m)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

Fext+vreldmdt=mdvdt\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; vrelv_{rel} is relative velocity of escaping mass.
dynamicsvariable_massjee-advanced
02

Applications of Newton's Laws

7 formulas

Atwood Machine Acceleration

a=m1−m2m1+m2ga = \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, m1>m2m_1 > m_2.
dynamicspulleysacceleration

Atwood Machine Tension

T=2m1m2m1+m2gT = \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

fcontact=m2Fm1+m2f_{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

∑F=F1+F2+F3=0\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

F=−kx\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=(m2+m3)Fm1+m2+m3T = \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=2mucos⁡θJ = 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
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03

Friction

6 formulas

Angle of Repose

θmax=tan⁡−1(μs)\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

fk=μkNf_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

(fs)max=μsN(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

fs≤μsNf_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⁡θ−μkcos⁡θ)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⁡θ+μkcos⁡θ)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
04

Dynamics of Circular Motion

6 formulas

Banked road speed

v0=Rgtan⁡θv_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

vmax=Rg(μs+tan⁡θ1−μstan⁡θ)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

fc=mv2R=mω2Rf_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

ω=gLcos⁡θ\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

vmax=μsRgv_{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

vbottom=5gRv_{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 ≥0\ge 0 at the top.
dynamicscircular_motionjee-advanced
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