How do I calculate projectile motion (range and height)?

Projectile motion formulas: Range R = v₀² × sin(2θ) / g. Maximum height H = v₀² × sin²(θ) / (2g). Time of flight T = 2v₀ × sin(θ) / g. Where v₀ is initial velocity (m/s), θ is launch angle (degrees), and g = 9.81 m/s². Air resistance is neglected in standard calculations.

What is the formula for the period of a simple pendulum?

Period T = 2π × √(L/g), where L is pendulum length (m) and g = 9.81 m/s² (gravitational acceleration). This approximation is valid for small angles (θ < 15°). For large angles, T = 2π × √(L/g) × (1 + θ²/16 + ...). Frequency f = 1/T Hz.

How do you calculate wave speed, frequency and wavelength?

Wave equation: v = f × λ, where v is wave speed (m/s), f is frequency (Hz), and λ is wavelength (m). For sound in air: v ≈ 343 m/s at 20°C. For electromagnetic waves: v = c = 3 × 10⁸ m/s. Energy of a photon: E = hf = hc/λ, where h = 6.626 × 10⁻³⁴ J·s (Planck's constant).

What is Snell's Law in optics?

Snell's Law: n₁ × sin(θ₁) = n₂ × sin(θ₂), where n₁ and n₂ are the refractive indices of the two media, and θ₁ and θ₂ are the angles of incidence and refraction measured from the normal. Total internal reflection occurs when θ₁ > θc = arcsin(n₂/n₁) for n₁ > n₂.

How do you calculate relativistic mass-energy equivalence?

Einstein's mass-energy equivalence: E = mc², where E is energy (Joules), m is mass (kg), and c = 2.998 × 10⁸ m/s. For moving objects, total energy E = γmc², where Lorentz factor γ = 1/√(1 - v²/c²). Kinetic energy KE = (γ - 1)mc². At v << c, this reduces to classical KE = ½mv².

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About Our Physics Calculators

Physics calculators on AI Calculator cover the equations students and engineers reach for most often: classical mechanics (kinematics, projectile motion, rotational motion), oscillations and waves (pendulum period, wave speed and frequency, Doppler shift), thermodynamics (ideal gas law, first-law energy balance), optics (Snell’s law, lens and mirror equations), electromagnetism (Coulomb’s law, Ohm’s law in physics form), and a small set of special relativity and quantum results (E = mc², de Broglie wavelength). Each tool shows the equation it is solving, the SI units of every input, and a worked example so the page also serves as a quick refresher when you have forgotten which constant goes where. The PDF report from each page includes the derivation step, so you can attach it to a problem set or a lab report.

Calculators in This Physics Suite

Projectile Motion Calculator — range, maximum height, time of flight, and trajectory for a projectile launched at angle θ with initial velocity v₀, neglecting air resistance.
Kinetic Energy Calculator — translational kinetic energy KE = ½mv² with optional rotational term ½Iω² for a rigid body.
Simple Pendulum Calculator — period T = 2π√(L/g) for the small-angle approximation, with a higher-order correction available for larger amplitudes.

Key Formulas at a Glance

For a projectile fired at angle θ with initial speed v₀ on a flat surface (g = 9.81 m/s²): range R = v₀² sin(2θ) / g, maximum height H = v₀² sin²(θ) / (2g), and time of flight T = 2v₀ sin(θ) / g. The maximum-range angle for level ground is 45°; for a launch above the landing point the optimal angle is less than 45° and the calculator lets you specify the launch height to handle that case.

For a simple pendulum of length L (m), the small-angle period is T = 2π √(L/g). This expression is good to within 1% for amplitudes up to about 15°. For larger amplitudes a better expression is T = 2π √(L/g) × (1 + θ₀²/16 + 11θ₀⁴/3072 + …), the leading terms of an elliptic-integral expansion.

Kinetic energy of a translating point mass is KE = ½mv². For a rigid body that is also rotating, the total kinetic energy is KE_total = ½mv_cm² + ½Iω², where I is the moment of inertia about the centre of mass and ω is the angular velocity. For a solid sphere I = (2/5)mr²; for a thin hoop I = mr²; for a uniform rod about its centre I = (1/12)mL².

Where These Calculators Help

For high-school and early-undergraduate physics, the projectile and pendulum calculators are useful for checking homework against an authoritative reference and for visualising how the answer changes with each input — try varying the launch angle from 30° to 60° on the projectile calculator and watch how the range curve peaks at 45°. For lab work, the pendulum calculator is a quick way to check whether your measured period matches the small-angle prediction, which is a common first lab in mechanics. For early-career engineers, the kinetic energy calculator is handy for sizing flywheels and rotating machinery (KE storage = ½Iω²; flywheel inertia from rim mass).

Frequently Asked Questions

Why is g taken as 9.81 m/s² and not 10?
Standard gravity at the Earth’s surface is g = 9.80665 m/s², usually rounded to 9.81 in engineering work. Using g = 10 introduces about 2% error, which is fine for back-of-envelope estimates but not for laboratory calculations. The calculators on this site let you set g manually if you are computing for a different latitude or a different planet.

Does the projectile calculator account for air resistance?
No — it solves the standard parabolic trajectory in vacuum. For real-world projectiles with significant drag (sports balls, ballistics), the trajectory deviates from the parabola and a numerical solver with drag coefficient is needed. The calculator will note when air drag becomes significant for the parameters you have entered.

What’s the difference between a simple and a physical pendulum?
A simple pendulum models the bob as a point mass at the end of a massless string. A physical pendulum has its mass distributed along its length and uses period T = 2π √(I / (mgd)), where d is the distance from the pivot to the centre of mass and I is the moment of inertia about the pivot.

Can these calculators be used for SI and CGS units?
The default is SI; the input fields support unit conversion to imperial (feet, ft/s, lbf) where appropriate. CGS (cm-gram-second) is not directly supported in the UI, but you can manually convert before entering.

Related calculator categories: Electrical engineering · Mechanical engineering · Chemical & process · Civil & structural · Engineering formulas reference