EQUATIONS

Gives the velocity of an object after it has been accelerating at a constant rate for a time t. Use

Gives the displacement of an object under constant acceleration after a time t. Use it when you need

Links initial velocity, final velocity, and displacement without requiring time. The equation is the

Gives the closing speed of two objects moving directly toward each other. The time to collision is t

Solves any equation of the form ax² + bx + c = 0 for x. In kinematics it appears whenever you set tw

Splits a launch velocity into independent horizontal and vertical components. Once decomposed, horiz

Gives the total airborne time for a projectile launched from and landing on the same horizontal leve

Gives the horizontal distance a projectile travels before returning to its launch height. Optimal ra

Gives the maximum height reached by a projectile above its launch point. At the peak, the vertical v

Three interrelated equations — v = v_0 + at, d = v_0·t + ½at², v² = v_0² + 2ad — collectively cover

Shows that two launch angles that sum to 90° — for example 30° and 60° — produce identical horizonta

Relates net force, mass, and acceleration. Given any two of the three quantities, the equation yield

Gives the gravitational force on a mass near Earth's surface. Weight is what a scale reads when the

Gives the maximum static friction or the kinetic friction force on an object in contact with a surfa

Gives the friction force on an already-sliding object. Once an object is in motion, kinetic friction

States that the net force on an object equals the vector sum of all individual forces acting on it.

States that every force has an equal and opposite reaction force on a different object. It is the re

Gives the viscous drag force on a slowly moving sphere in a fluid (Stokes drag): F = bv, where b = 6

Gives the aerodynamic drag on a body moving at moderate to high speeds: F = ½ρC_d Av². The drag grow

Gives the constant (maximum) falling speed reached when drag exactly balances gravity. For linear dr

Gives the oscillation period of a simple pendulum when the swing amplitude is small (θ₀ ≲ 15°). The

Gives the angular frequency ω = √(g/L) of a small-angle pendulum. Knowing ω, you can write x(t) = A·

Converts the potential energy at the top of the swing (mgh) into kinetic energy at the bottom (½mv²)

Extends the small-angle pendulum period to larger amplitudes by adding successive correction terms:

Gives the exact period of a simple pendulum at any amplitude via the complete elliptic integral of t

Defines K(k) = ∫₀^{π/2} dφ / √(1 − k²sin²φ), a definite integral that appears in the exact large-ang

Expresses the exact elliptic-integral period as a power series in θ₀: T = T₀·Σ [(2n)!/(2²ⁿ(n!)²)]²·θ

The universal differential equation x″ + ω²x = 0 describes any system with a linear restoring force.

Gives the natural angular frequency of a mass–spring system: ω = √(k/m). From ω you get period T = 2

Gives the constant total mechanical energy of an undamped oscillator: E = ½kA². Energy oscillates be

Gives the displacement of a simple harmonic oscillator at any time t: x(t) = A·cos(ωt + φ). The ampl

Gives the velocity of a simple harmonic oscillator at any time t: v(t) = −Aω·sin(ωt + φ). It is the

Gives the acceleration of a simple harmonic oscillator: a(t) = −Aω²·cos(ωt + φ) = −ω²·x(t). Accelera

Gives the two normal-mode frequencies of two identical pendulums coupled by a weak spring: ω₁ = ω₀ (

Gives the position of an underdamped oscillator: x(t) = A₀·e^(−γt/2)·cos(ω_d·t + φ). The exponential

Gives the amplitude envelope of a damped oscillator as a function of time: A(t) = A₀·e^(−γt/2). The

Q = ω₀/γ characterizes how sharply an oscillator resonates. A high Q means narrow bandwidth, slow en

Gives the amplitude of the steady-state response when a force F₀·cos(ω_d·t) drives a damped oscillat

States that maximum steady-state amplitude occurs when the driving frequency ω_d is close to the nat

Gives the actual oscillation frequency of an underdamped system: ω_d = √(ω₀² − γ²/4). Damping always