Key Kinematics Formulas and Motion Equations
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Kinematics Formulas and Motion Equations
Vector Description of Motion
| Position, Velocity, and Acceleration Vectors | ||
| r(t) | v(t) = dr(t) / dt | a(t) = dv(t) / dt |
| a(t) | v(t) = ∫ a(t) dt + constant | r(t) = ∫ v(t) dt + constant |
| Intrinsic Components of Acceleration: a = at + an | ||
| T: Unit vector tangent to the trajectory. T = v / |v| | ||
| N: Normal unit vector to the trajectory. | ||
| at = Tangential acceleration; at = dv / dt | ||
| an = Normal acceleration = v2 / ρ; ρ is the radius of curvature. | ||
Rectilinear Motion
| Uniform Rectilinear Motion (a = 0) | ||
| a = 0 | v = v0 = constant | x = x0 + v0t |
| Uniformly Accelerated Rectilinear Motion (a = constant) | ||
| a = cte | v = v0 + at | x = x0 + v0t + at2 / 2 |
| v2 - v02 = 2a(x - x0) | ||
Circular Motion
| General Circular Motion | ||
| θ (Angular Position) | ω = dθ / dt (Angular Velocity) | α = dω / dt (Angular Acceleration) |
| s = θR (Arc Length) | v = ωR (Tangential Speed) | at = Rα an = ω2R |
| ω (Angular frequency), n (frequency), T (Period) | ω = 2πn = 2π / T | |
| Uniform Circular Motion (α = 0) | ||
| α = 0 | ω = ω0 = cte | θ = θ0 + ωt |
| Uniformly Accelerated Circular Motion (ω = constant) | ||
| α = cte | ω = ω0 + αt | θ = θ0 + ω0t + αt2 / 2 |
Simple Harmonic Motion (SHM)
| SHM Characteristics | ||
| Differential Equation | d2x(t) / dt2 + ω2x = 0 | ω = 2πn = 2π / T |
| ω (Angular frequency), n (frequency), T (Period) | ||
| Solution | x(t) = A cos(ωt + φ) | A: Amplitude φ: Initial Phase |
| Period of a Spring | T = 2π √m / k | k: Spring constant |
| Period of a Simple Pendulum | T = 2π √l / g | l: Length of the string |
| Period of a Physical Pendulum | T = 2π √I / mgl | I: Moment of inertia |
| Period of a Torsion Pendulum | T = 2π √I / κ | |
Projectile Motion (Parabolic Trajectory)
| Launch with initial velocity v0 and angle θ | ||
| Horizontal (ax = 0) | vx = v0 cos(θ) = constant | x = v0 cos(θ)t |
| Vertical (ay = -g) | vy = v0 sin(θ) - gt | y = v0 sin(θ)t - gt2 / 2 |
| Maximum Range: Rmax = v02 sin(2θ) / g | ||
| Maximum Height: Hmax = v02 sin2(θ) / (2g) | ||
Kinematics in Polar Coordinates
| Polar Coordinates | |
| Position Vector | r = r ur |
| Velocity | v = (dr/dt) ur + (r dθ/dt) uθ |
| Acceleration | a = [d2r/dt2 - r(dθ/dt)2]ur + [2(dr/dt)(dθ/dt) + r(d2θ/dt2)]uθ |
Kinematics of Relative Motion
| Relative Motion Formulas | |
| OXYZ (Inertial Frame). Lowercase: quantities relative to the Inertial Frame (SRI). Lowercase with prime: quantities relative to the Non-Inertial Frame (SRNI). Capital letters: quantities related to the origin of the SRNI in the SRI. | |
| Position Vector | r = R + r' |
| Velocity | v = V + ω x r' + v' |
| Drift Velocity: va = V + ω x r' | |
| Acceleration | a = A + α x r' + ω x (ω x r') + 2ω x v' + a' |
| Transport Acceleration: at = A + α x r' + ω x (ω x r') | |
| Coriolis Acceleration: ac = 2ω x v' | |
International System (SI) Units
| Fundamental Units and Dimensions | ||
| Time | second (s) | T |
| Position (Space) | meter (m) | L |
| Speed | m / s | LT-1 |
| Acceleration | m / s2 | LT-2 |
| Angular Position | radian (rad) | |
| Angular Velocity | rad / s | |
| Angular Acceleration | rad / s2 | |