Mechanics Formulas

Complete collection of mechanics formulas with detailed explanations, notation meanings, units, and real-world applications. Master the fundamentals of motion and forces.

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Newton's Laws of Motion

Newton's First Law (Law of Inertia)

\sum F = 0 \rightarrow a = 0

An object at rest stays at rest, and an object in motion stays in motion unless acted upon by a net external force.

Notation:

0

:Zero (no change in motion)

\sum F

:Net force (vector sum of all forces)

a

:Acceleration

Units:

F

:Newton (N) = kg·m/s²

a

:m/s²

Applications:

•Objects at rest on a table
•Satellites in orbit
•Car continuing to move after engine stops

Limitations:

Only applies in inertial reference frames

Newton's Second Law

F = ma

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

Notation:

F

:Net force (vector)

m

:Mass of the object

a

:Acceleration (vector)

Units:

F

:Newton (N)

m

:Kilogram (kg)

a

:m/s²

Applications:

•Rocket propulsion
•Car acceleration
•Falling objects

Limitations:

Valid for speeds much less than speed of light

Newton's Third Law

F_{12} = -F_{21}

For every action, there is an equal and opposite reaction.

Notation:

F_{12}

:Force exerted by object 1 on object 2

F_{21}

:Force exerted by object 2 on object 1

-

:Negative sign indicates opposite direction

Units:

F_{12}, F_{21}

:Newton (N)

Applications:

•Rocket propulsion
•Walking
•Swimming

Limitations:

Forces act on different objects

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Kinematics Equations

First Equation of Motion

v = v_0 + at

Final velocity equals initial velocity plus acceleration times time.

Notation:

v

:Final velocity

v_0

:Initial velocity

a

:Acceleration

t

:Time

Units:

v, v_0

:m/s

a

:m/s²

t

Applications:

•Car acceleration
•Free fall
•Projectile motion

Limitations:

Constant acceleration only

Second Equation of Motion

s = v_0t + \frac{1}{2}at^2

Displacement equals initial velocity times time plus half acceleration times time squared.

Notation:

s

:Displacement

v_0

:Initial velocity

t

:Time

a

:Acceleration

Units:

s

v_0

:m/s

t

a

:m/s²

Applications:

•Distance traveled by car
•Height of falling object
•Range of projectile

Limitations:

Constant acceleration, one-dimensional motion

Third Equation of Motion

v^2 = v_0^2 + 2as

Final velocity squared equals initial velocity squared plus twice acceleration times displacement.

Notation:

v

:Final velocity

v_0

:Initial velocity

a

:Acceleration

s

:Displacement

Units:

v, v_0

:m/s

a

:m/s²

s

Applications:

•Braking distance
•Escape velocity
•Impact velocity

Limitations:

Constant acceleration, one-dimensional motion

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Work and Energy

Work Done

W = F \cdot s = Fs \cos \theta

Work equals force times displacement times cosine of angle between them.

Notation:

W

:Work done

F

:Force

s

:Displacement

\theta

:Angle between force and displacement

Units:

W

:Joule (J) = N·m

F

s

\theta

:radians or degrees

Applications:

•Lifting objects
•Pushing a cart
•Spring compression

Limitations:

Force must be constant

Kinetic Energy

KE = \frac{1}{2}mv^2

Kinetic energy equals half mass times velocity squared.

Notation:

KE

:Kinetic energy

m

:Mass

v

:Velocity

Units:

KE

:Joule (J)

m

:kg

v

:m/s

Applications:

•Moving vehicles
•Falling objects
•Collision analysis

Limitations:

Non-relativistic speeds

Gravitational Potential Energy

PE = mgh

Gravitational potential energy equals mass times gravitational acceleration times height.

Notation:

PE

:Potential energy

m

:Mass

g

:Gravitational acceleration

h

:Height

Units:

PE

:Joule (J)

m

:kg

g

:m/s²

h

Applications:

•Water in dam
•Roller coaster
•Falling objects

Limitations:

Near Earth's surface, constant g

Conservation of Mechanical Energy

KE + PE = \text{constant}

Total mechanical energy (kinetic plus potential) remains constant in the absence of non-conservative forces.

Notation:

KE

:Kinetic energy

PE

:Potential energy

\text{constant}

:Total mechanical energy

Units:

KE, PE, \text{constant}

:Joule (J)

Applications:

•Pendulum motion
•Roller coaster
•Spring-mass system

Limitations:

Only conservative forces present

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Momentum and Collisions

Linear Momentum

p = mv

Momentum equals mass times velocity.

Notation:

p

:Momentum (vector)

m

:Mass

v

:Velocity (vector)

Units:

p

:kg·m/s

m

:kg

v

:m/s

Applications:

•Collision analysis
•Rocket propulsion
•Sports physics

Limitations:

Non-relativistic speeds

Conservation of Momentum

\sum p_{\text{initial}} = \sum p_{\text{final}}

Total momentum of a system remains constant if no external forces act on it.

Notation:

\sum p_{\text{initial}}

:Sum of initial momenta

\sum p_{\text{final}}

:Sum of final momenta

Units:

\sum p_{\text{initial}}, \sum p_{\text{final}}

:kg·m/s

Applications:

•Elastic collisions
•Inelastic collisions
•Explosions

Limitations:

No external forces on system

Impulse

J = F\Delta t = \Delta p

Impulse equals force times time interval, which equals change in momentum.

Notation:

J

:Impulse

F

:Average force

\Delta t

:Time interval

\Delta p

:Change in momentum

Units:

J

:N·s = kg·m/s

F

\Delta t

\Delta p

:kg·m/s

Applications:

•Car crashes
•Baseball hits
•Airbag deployment

Limitations:

Constant force approximation

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Circular Motion

Centripetal Force

F_c = \frac{mv^2}{r}

Centripetal force equals mass times velocity squared divided by radius.

Notation:

F_c

:Centripetal force

m

:Mass

v

:Tangential velocity

r

:Radius of circular path

Units:

F_c

m

:kg

v

:m/s

r

Applications:

•Car turning
•Satellite orbits
•Amusement park rides

Limitations:

Uniform circular motion

Angular Velocity

\omega = \frac{v}{r} = 2\pi f

Angular velocity equals tangential velocity divided by radius, or 2π times frequency.

Notation:

\omega

:Angular velocity

v

:Tangential velocity

r

:Radius

f

:Frequency

Units:

\omega

:rad/s

v

:m/s

r

f

:Hz

Applications:

•Rotating objects
•Planetary motion
•Centrifuges

Limitations:

Uniform circular motion

Centripetal Acceleration

a_c = \frac{v^2}{r} = \omega^2 r

Centripetal acceleration equals velocity squared divided by radius, or angular velocity squared times radius.

Notation:

a_c

:Centripetal acceleration

v

:Tangential velocity

r

:Radius

\omega

:Angular velocity

Units:

a_c

:m/s²

v

:m/s

r

\omega

:rad/s

Applications:

•Banked curves
•Artificial gravity
•Centrifugal separators

Limitations:

Uniform circular motion

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Gravitation

Universal Law of Gravitation

F = G\frac{m_1 m_2}{r^2}

Gravitational force between two masses equals gravitational constant times product of masses divided by distance squared.

Notation:

F

:Gravitational force

G

:Gravitational constant

m_1, m_2

:Masses of objects

r

:Distance between centers

Units:

F

G

:6.674 × 10⁻¹¹ N·m²/kg²

m_1, m_2

:kg

r

Applications:

•Planetary motion
•Satellite orbits
•Tides

Limitations:

Point masses or spherical objects

Gravitational Field Strength

g = \frac{GM}{r^2}

Gravitational field strength equals gravitational constant times mass divided by distance squared.

Notation:

g

:Gravitational field strength

G

:Gravitational constant

M

:Mass of central body

r

:Distance from center

Units:

g

:m/s²

G

:N·m²/kg²

M

:kg

r

Applications:

•Weight on different planets
•Satellite motion
•Escape velocity

Limitations:

Spherical mass distribution

Escape Velocity

v_{\text{escape}} = \sqrt{\frac{2GM}{r}}

Escape velocity equals square root of twice gravitational constant times mass divided by radius.

Notation:

v_{\text{escape}}

:Escape velocity

G

:Gravitational constant

M

:Mass of central body

r

:Radius from center

Units:

v_{\text{escape}}

:m/s

G

:N·m²/kg²

M

:kg

r

Applications:

•Rocket launches
•Black hole physics
•Space exploration

Limitations:

No air resistance, spherical body

Practice Problems

📝Calculate force needed to accelerate 2 kg mass at 3 m/s²
📝Find kinetic energy of 5 kg object moving at 10 m/s
📝Calculate centripetal force for 1 kg mass in 2m radius at 5 m/s

Try Interactive Calculators

Projectile Motion Newton's Laws Circular Motion Work & Energy Simple Harmonic Motion Conservation of Momentum Elastic Collision Centripetal Force

Study Tips

💡Always check units for consistency
💡Draw free-body diagrams for force problems
💡Use energy conservation when possible