Physics Formulas & Calculations: A Practical Reference Guide

Classical physics describes the behaviour of objects at everyday speeds and scales using a relatively small set of equations. Whether you are a student working through a problem set, an engineer doing a back-of-the-envelope check, or simply curious about how the physical world works, having these formulas close at hand—along with worked examples—makes the subject far more approachable. This guide covers the core equations of mechanics, waves, and nuclear physics, with links to calculators that do the arithmetic for you.

Newton's Second Law: Force, Mass, and Acceleration

The most fundamental equation in classical mechanics states that the net force on an object equals its mass multiplied by its acceleration:

F = m × a

Quantity	Symbol	SI Unit
Force	F	Newton (N)
Mass	m	Kilogram (kg)
Acceleration	a	m/s²

A 1,500 kg car accelerating at 3 m/s² requires a net force of 4,500 N. The Force Calculator (F = ma) solves for any of the three variables when the other two are known.

Kinetic and Potential Energy

Energy exists in two fundamental mechanical forms. Kinetic energy (KE) is the energy of motion:

KE = ½ × m × v²

A 70 kg cyclist moving at 10 m/s (36 km/h) has KE = 0.5 × 70 × 100 = 3,500 J. Use the Kinetic Energy Calculator for any mass-velocity combination. Gravitational potential energy (PE) depends on mass, gravitational acceleration, and height:

PE = m × g × h

Where g = 9.81 m/s² at Earth’s surface. That same 70 kg cyclist at the top of a 50 m hill has PE = 70 × 9.81 × 50 = 34,335 J. The Potential Energy Calculator (PE = mgh) handles unit conversions between Joules and kilojoules.

Work and Power

Work is the energy transferred when a force moves an object through a distance:

W = F × d × cos(θ)

The cosine term accounts for the angle between the force vector and the direction of motion. If the force is parallel to motion, θ = 0 and cos(0) = 1, so W = F × d. Calculate work for any force-distance-angle combination with the Work Calculator (W = Fd·cosθ).

Power is the rate of doing work:

P = W / t

Lifting 100 kg through 2 m in 4 seconds involves work of (100 × 9.81 × 2) = 1,962 J and power of 1,962 / 4 = 490.5 W. The Power Calculator (P = W/t) solves for power, work, or time.

Kinematics: Speed, Distance, and Time

At constant velocity, the relationship between speed, distance, and time is:

Distance = Speed × Time

The Speed, Distance & Time Calculator solves for any of the three variables and supports unit conversions between mph, km/h, m/s, and more. It also handles problems involving acceleration using the kinematic equations (v = u + at, s = ut + ½at²).

Projectile Motion

A projectile launched at angle θ with initial velocity v⊂0; follows a parabolic path. The key equations are:

Range: R = (v⊂0;² × sin(2θ)) / g
Max height: H = (v⊂0;² × sin²(θ)) / (2g)
Time of flight: T = (2 × v⊂0; × sin(θ)) / g

At 45°, range is maximized. A ball launched at 20 m/s at 45° travels (400 × sin 90°) / 9.81 = 40.8 m. The Projectile Motion Calculator plots the full trajectory and computes all three quantities for any launch angle and speed.

Waves: Frequency, Wavelength, and the Speed of Light

For any wave, the relationship between wave speed (c), frequency (f), and wavelength (λ) is:

c = f × λ

For electromagnetic waves in a vacuum, c = 3 × 10&sup8; m/s. A radio station broadcasting at 100 MHz has a wavelength of (3 × 10&sup8;) / (10&sup8;) = 3 m. Calculate any of the three quantities with the Wavelength & Frequency Calculator (c = fλ). For audio and digital signals, the Sample Rate & Nyquist Calculator applies the Nyquist theorem (minimum sample rate must be at least twice the highest signal frequency) to digital sampling problems.

Terminal and Escape Velocity

Terminal velocity is reached when drag force equals gravitational force, so net acceleration becomes zero:

v⊂t; = √(2mg / (ρ × C⊂D; × A))

Where ρ is air density, C⊂D; is drag coefficient, and A is cross-sectional area. A skydiver in a spread-eagle position has a terminal velocity of roughly 55 m/s (195 km/h). The Terminal Velocity Calculator handles all inputs. Escape velocity is the minimum speed needed to break free of a planet’s gravitational field without further propulsion:

v⊂e; = √(2GM / r)

Earth’s escape velocity is approximately 11.2 km/s. The Escape Velocity Calculator computes this for any planetary mass and radius.

Radioactive Decay and Half-Life

Radioactive decay follows an exponential model. The half-life (t½) is the time for half of the remaining atoms to decay:

N(t) = N⊂0; × (1/2)^(t / t½)

After 3 half-lives, (1/2)³ = 1/8 of the original material remains. The Half-Life Calculator solves for remaining quantity, elapsed time, or number of half-lives and includes a preset list of common isotopes with their known half-lives.

Common Mistakes to Avoid

Forgetting to convert units: Mixed units are the most common source of error in physics calculations. Always convert to SI (meters, kilograms, seconds) before substituting into formulas.
Ignoring the angle in work calculations: If a force is applied at an angle to the direction of motion, only the parallel component does work. Forgetting the cos(θ) term overstates the work done.
Assuming terminal velocity is constant for all objects: Terminal velocity depends on mass, drag coefficient, cross-sectional area, and fluid density. A feather and a bowling ball have very different terminal velocities in air.
Confusing frequency and angular frequency: ω = 2πf. When using equations from wave mechanics, check whether the formula expects frequency (Hz) or angular frequency (rad/s).

Frequently Asked Questions

What is the difference between speed and velocity?

Speed is a scalar quantity (magnitude only), while velocity is a vector (magnitude and direction). An object traveling in a circle at constant speed has changing velocity because its direction is continuously changing—meaning it is accelerating even though its speed is constant.

How does air density affect terminal velocity?

Terminal velocity is inversely proportional to the square root of air density. At high altitude where air is thinner, drag is lower and terminal velocity is higher. Skydivers jumping from high altitude reach higher speeds than at sea level. The Terminal Velocity Calculator lets you adjust air density for altitude.

What is a sine wave and when is it used?

A sine wave is a smooth, periodic oscillation described by the function A×sin(2πft + φ), where A is amplitude, f is frequency, and φ is phase. Sine waves model sound, alternating current, electromagnetic radiation, and many mechanical oscillations. Use the Sine Wave Calculator & Plotter to visualise any sine wave and read off values at specific time points.