Physics Formulas Guide

Kinematics Equations

Kinematics describes motion without considering its causes. These equations relate displacement, velocity, acceleration, and time under constant acceleration.

v = v₀ + at

Final velocity from initial velocity and acceleration

x = v₀t + ½at²

Displacement with constant acceleration

v² = v₀² + 2ax

Velocity-displacement relationship (no time)

x = ½(v₀ + v)t

Displacement with average velocity

Derivation: x = v₀t + ½at²

Start with: Acceleration is the rate of change of velocity: a = dv/dt

Integrate: v = ∫a dt = at + C�? where C�?= v₀ (initial velocity)

So: v = v₀ + at

Now: Velocity is the rate of change of displacement: v = dx/dt

Integrate: x = ∫v dt = �?v₀ + at) dt = v₀t + ½at² + C�?/span>

Result: x = v₀t + ½at² (assuming x₀ = 0)

Newton's Laws of Motion

Newton's three laws form the foundation of classical mechanics, describing how forces affect motion.

1st Law

Law of Inertia

An object remains at rest or in uniform motion unless acted upon by a net external force.

Implication: If ΣF = 0, then a = 0 (no acceleration)

2nd Law

Force = Mass × Acceleration

F = ma

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

3rd Law

Action-Reaction

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

Implication: F₁₂ = -F₂₁

Free Body Diagrams

To apply Newton's laws, identify all forces acting on an object:

Weight (W = mg): Always points downward
Normal Force (N): Perpendicular to contact surface
Friction (f = μN): Opposes motion direction
Tension (T): Along ropes/strings, away from object
Applied Force (F): External push/pull

Work & Energy

Energy is the capacity to do work. The work-energy theorem connects force, displacement, and changes in kinetic energy.

Key Energy Formulas

Work

W = F · d · cos(θ)

Kinetic Energy

KE = ½mv²

Potential Energy

PE = mgh

Work-Energy Theorem

W = ΔKE = ½mv² - ½mv₀²

Derivation: Kinetic Energy Formula

Start with: Work = Force × Displacement: W = Fd

Newton's 2nd: F = ma, so W = mad

From kinematics: v² = v₀² + 2ad, so ad = (v² - v₀²)/2

Substitute: W = m(v² - v₀²)/2 = ½mv² - ½mv₀²

Insight: KE = ½mv² is the kinetic energy at velocity v

Conservation of Energy

KE�?+ PE�?= KE�?+ PE�?/p>

In a closed system with no non-conservative forces, total mechanical energy remains constant.

Momentum & Collisions

Momentum is "mass in motion" and is always conserved in isolated systems, making it invaluable for analyzing collisions.

Momentum Formulas

Momentum

p = mv

Impulse

J = FΔt = Δp

Conservation

m₁v�?+ m₂v�?= m₁v�? + m₂v�?

Elastic Collision

Both momentum AND kinetic energy are conserved.

Objects bounce apart
No energy lost to deformation
Example: Billiard balls

Inelastic Collision

Only momentum is conserved; kinetic energy is lost.

Objects may stick together
Energy converts to heat/deformation
Example: Car crashes

Circular Motion

Objects moving in circles require a centripetal (center-seeking) force to maintain their curved path.

Circular Motion Formulas

Centripetal Acceleration

a꜀ = v²/r = ω²r

Centripetal Force

F꜀ = mv²/r = mω²r

Period

T = 2πr/v = 2π/ω

Common Misconception: "Centrifugal force" is not a real force—it's the sensation of inertia trying to keep you moving in a straight line while the car/seat exerts centripetal force to curve your path.

Gravitation

Newton's Law of Universal Gravitation describes the attractive force between any two masses in the universe.

Newton's Law of Universal Gravitation

F = G(m₁m�?/r²

G = 6.674 × 10⁻¹�?N·m²/kg² r = distance between centers

Derivation: g = 9.8 m/s² at Earth's Surface

From F = ma: Weight = mg (where g is gravitational acceleration)

From Universal Gravitation: F = GMm/R² (M = Earth's mass, R = Earth's radius)

Set equal: mg = GMm/R², so g = GM/R²

Calculate: g = (6.674×10⁻¹�?(5.972×10²�?/(6.371×10�?²

Result: g �?9.81 m/s²