Kinematics — Describing Motion
Kinematics is the mathematical description of motion without asking what causes it. Before we can understand forces, we need a precise language for displacement, velocity, and acceleration.
Displacement, Velocity, and Acceleration
Displacement (s) is the change in position — a vector quantity with both magnitude and direction. It is not the same as distance: a runner who completes one full lap of a track has zero displacement but non-zero distance.
Velocity (v) is the rate of change of displacement — how fast position is changing and in which direction. Average velocity = total displacement ÷ time elapsed. Instantaneous velocity is the limit of this as the time interval approaches zero.
Acceleration (a) is the rate of change of velocity. An object accelerates whenever its speed changes, its direction changes, or both. Crucially, an object moving at constant speed around a circle is always accelerating — its direction is changing continuously.
Speed is the magnitude of velocity. Velocity is a vector; speed is a scalar. A car braking at 30 m/s to 20 m/s has the same speed direction but a negative acceleration (deceleration). A car turning a corner at constant speed has zero speed change but nonzero acceleration — it's changing direction.
The SUVAT Equations (Uniform Acceleration)
When acceleration is constant, five variables describe the motion completely: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). Four equations connect them — each one missing one variable.
| Equation | Missing Variable | When to Use |
|---|---|---|
| v = u + at | s (displacement) | When you know u, a, t and need v |
| s = ut + ½at² | v (final velocity) | When you know u, a, t and need s |
| v² = u² + 2as | t (time) | When time is unknown or not needed |
| s = ½(u + v)t | a (acceleration) | When acceleration is unknown |
| s = vt − ½at² | u (initial velocity) | When initial velocity is unknown |
Projectile Motion
A projectile moves under gravity alone — no air resistance. The key insight is that horizontal and vertical motions are completely independent. Horizontal: constant velocity (no horizontal force). Vertical: constant downward acceleration g = 9.8 m/s².
Horizontal Motion
No horizontal force, so no horizontal acceleration. The horizontal velocity never changes throughout the flight.
x = v₀cosθ · tVertical Motion
Constant downward acceleration g. Vertical velocity starts at v₀sinθ and decreases to zero at maximum height, then increases downward.
y = v₀sinθ · t − ½gt²Maximum Range
Range is maximised at 45° launch angle — this is where horizontal and vertical velocity components are equal, optimally sharing the total initial speed.
R = v₀²sin2θ / gMaximum Height
At maximum height, the vertical velocity is exactly zero. Only horizontal motion remains at the apex of the trajectory.
H = v₀²sin²θ / 2gNewton's Three Laws of Motion
Published in the Principia Mathematica in 1687, Newton's three laws remain the foundation of classical mechanics — and of every branch of engineering built upon it. Understanding them deeply, not just memorising them, is essential.
An object at rest stays at rest, and an object in motion stays in motion at constant velocity, unless acted upon by a net external force.
This overthrew Aristotle's 2,000-year-old idea that objects naturally come to rest. The natural state is constant velocity — including zero. Inertia is the tendency of matter to resist changes to its state of motion. Mass is the measure of inertia.
"Moving objects need a force to keep moving." — False. A hockey puck sliding on frictionless ice continues at constant velocity indefinitely. Friction is a real force that decelerates objects on Earth — remove it and no driving force is needed to maintain constant speed.
The net force on an object equals its mass times its acceleration: F = ma
This is the equation of motion — the most practically powerful of the three laws. Net force and acceleration are always vectors pointing in the same direction. Mass is the proportionality constant: doubling mass halves acceleration for the same force.
For every action there is an equal and opposite reaction — forces always come in pairs acting on different objects.
The critical phrase is "on different objects." Action-reaction pairs can never cancel each other because they act on separate bodies. A rocket pushes exhaust backward; the exhaust pushes the rocket forward. The rocket accelerates because only the forward force acts on it.
Rocket Propulsion
Exhaust gases pushed backward → reaction force pushes rocket forward. The paired forces act on different objects (rocket vs. gases).
Walking
Foot pushes backward on ground → ground pushes foot (and you) forward. Without this reaction force you couldn't move forward.
Swimming
Hand pushes water backward → water pushes swimmer forward. The same principle governs fish fins, bird wings, and boat propellers.
Energy, Work, and Power
Energy is the capacity to do work. It cannot be created or destroyed — only converted between forms. The work-energy theorem connects forces directly to changes in kinetic energy, making it one of the most powerful tools in mechanics.
Work and the Work-Energy Theorem
Work is done when a force causes displacement in the direction of the force. If a force F acts at angle θ to the direction of motion over displacement s, the work done is W = Fs·cosθ. Work is a scalar — it has no direction, only magnitude and sign.
The Work-Energy Theorem states that the net work done on an object equals its change in kinetic energy: W_net = ΔKE = ½mv² − ½mu². This is one of the most powerful shortcuts in mechanics — it bypasses the need to know forces at every instant.
Kinetic Energy
Energy of motion. Depends on mass and the square of velocity — doubling speed quadruples kinetic energy.
KE = ½mv²Gravitational PE
Energy stored by an object due to its height above a reference level. Depends on mass, gravity, and height.
GPE = mghElastic PE
Energy stored in a compressed or stretched spring. Depends on the spring constant and deformation squared.
EPE = ½kx²Power
Rate of doing work or transferring energy. Measured in watts (W = J/s). Also equals force × velocity for constant motion.
P = W/t = FvConservation of Energy
In an isolated system with no non-conservative forces (like friction), total mechanical energy is conserved: KE + PE = constant. When friction acts, mechanical energy is not conserved — but total energy (including heat) still is. This is always true — it's one of the deepest laws in physics.
The total energy of an isolated system never changes. Energy converts between kinetic, potential, thermal, and other forms — but the total is constant. This law has never been found to be violated in any experiment.
Momentum and Collisions
Momentum is the "quantity of motion" — a vector that combines both mass and velocity. Its conservation is one of the most powerful principles in physics, applying even when energy is not conserved.
Elastic Collision
Both momentum AND kinetic energy are conserved. Ideal billiard balls approximate this. Particles bounce off each other without energy loss.
Inelastic Collision
Momentum is conserved but kinetic energy is not — some converts to heat or deformation. Real-world collisions are inelastic.
Perfectly Inelastic
Objects stick together after collision — maximum kinetic energy loss while momentum is still conserved. Car crashes, clay impacts.
Circular Motion
An object moving in a circle at constant speed is constantly accelerating — because its direction is constantly changing. This centripetal acceleration always points toward the centre of the circle and requires a centripetal force to maintain.
"Centrifugal force" is not a real force — it is a fictitious force felt in a rotating reference frame. In an inertial frame, there is only centripetal force pulling inward. When a car turns a corner and you feel pushed outward, your body's inertia is resisting the inward centripetal force — you are not being pushed outward by any real force.
Planetary Orbits
Gravity provides the centripetal force for orbital motion. For a circular orbit: GMm/r² = mv²/r, giving orbital speed v = √(GM/r).
Vertical Circles
At the top of a loop, gravity and normal force both point down (toward centre). At the bottom, normal force points up and gravity down — the difference provides centripetal force.
Banked Curves
Roads banked at angle θ allow cars to navigate curves without friction — the horizontal component of the normal force provides centripetal force: tanθ = v²/rg.
Universal Gravitation
Newton's Law of Universal Gravitation describes the attractive force between any two masses — from falling apples to orbiting planets. It was the first unified physical law, explaining both terrestrial and celestial mechanics with one equation.
The inverse-square relationship (1/r²) means that doubling the distance reduces the gravitational force by a factor of 4. This law applies at any scale — from a person on Earth to galaxies attracting each other across millions of light-years.
Gravitational Field Strength
The gravitational field strength g at any point is the force per unit mass experienced by a test mass at that point: g = GM/r². At Earth's surface (r = 6.37 × 10⁶ m), this gives g ≈ 9.8 N/kg = 9.8 m/s². The field decreases with altitude — astronauts on the ISS (400 km altitude) still experience about 89% of surface gravity, which is why they fall freely around Earth rather than floating weightlessly in zero gravity.
Rotational Dynamics
Rotational dynamics is the angular equivalent of Newton's laws — every concept from linear mechanics has a rotational counterpart. Understanding the parallels makes both systems far easier to master.
| Linear Concept | Linear Symbol | Rotational Analogue | Rotational Symbol |
|---|---|---|---|
| Displacement | s | Angular displacement | θ (rad) |
| Velocity | v | Angular velocity | ω (rad/s) |
| Acceleration | a | Angular acceleration | α (rad/s²) |
| Mass | m | Moment of inertia | I (kg·m²) |
| Force | F | Torque | τ (N·m) |
| F = ma | τ = Iα | ||
| Momentum p = mv | Angular momentum L = Iω | ||
| KE = ½mv² | KE_rot = ½Iω² |
In the absence of external torques, the angular momentum L = Iω of a system is conserved. This is why a spinning figure skater spins faster when pulling their arms in — reducing r reduces I, and since L = Iω must be constant, ω must increase.
Classical Mechanics Formula Sheet
Every essential equation for classical mechanics, organized by topic. Bookmark this page as your go-to reference.
Kinematics
Newton's Laws & Forces
Energy & Momentum
Rotational Dynamics
Practice Problems
Test your understanding. Try each problem before revealing the solution — the struggle before the answer is where real learning happens.
Horizontal distance: x = v_x × t = 12 × 3 = 36 m from the base.
Friction force: f = μN = 0.3 × 19.6 = 5.88 N
Net force: F_net = 10 − 5.88 = 4.12 N
Acceleration: a = F/m = 4.12/2 = 2.06 m/s²
(60)(4) + (20)(0) = (80)v
240 = 80v → v = 3 m/s
Tension (provides centripetal force): T = ma_c = 0.3 × 31.25 = 9.375 N
½(400)(0.15)² = ½(0.1)v²
4.5 = 0.05v² → v² = 90 → v = √90 ≈ 9.49 m/s
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Frequently Asked Questions
Second Law: F = ma — the net force on an object equals its mass times its acceleration. The most useful law for solving mechanics problems.
Third Law: Every action has an equal and opposite reaction — forces come in pairs acting on different objects. A book on a table pushes the table down; the table pushes the book up with equal force.
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