Kinematics Formulas: Motion, Speed, and Acceleration

Kinematics Formulas

Position Vector

r = xi + yj

• x = r cos
• y = r sin
• r = √(x² + y²)
• tan θ = y / x

Displacement

Δr = r - r_initial

Speed, Average Speed, Instantaneous Speed

• Average Speed: v_av = Δr / Δt
• Instantaneous Speed: v = dr / dt

Average Acceleration, Instantaneous Acceleration

• Average Acceleration: a_av = Δv / Δt
• Instantaneous Acceleration: a = dv / dt

Uniform Rectilinear Motion (MRU)

• v = Δx / Δt
• v_mean = (v₀ + v) / 2
• v = v₀ + at
• x = x₀ + vt
• x = x₀ + v₀t + (1/2)at²
• v² - v₀² = 2aΔx
• v² = v₀² ± 2as

Free Fall

• Velocity: v = gt
• Position (height fallen): y = (1/2)gt²
• Velocity (upward): v = -gt
• Position (height): y = y₀ - (1/2)gt²

Upward Vertical Launch

• Velocity: v = v₀ - gt
• Position (height): y = y₀ + v₀t - (1/2)gt²
• Time to reach maximum height: t = v₀ / g
• Maximum height: y_max = v₀² / (2g)
• Time of flight: t_flight = 2v₀ / g

Horizontal Launch

• Horizontal component (MRU): x = v₀t
• Vertical component (MRU): y = y₀ - (1/2)gt²
• Position: r = v₀ti + (y₀ - (1/2)gt²)j
• Velocity: v = v₀i - gtj
• v = √(v_x² + v_y²)
• Horizontal velocity: v_x = v₀
• Vertical velocity: v_y = -gt

Parabolic Motion

• v_0x = v₀cos θ
• v_0y = v₀sin θ
• y = y₀ + v_0yt - (1/2)gt²
• Horizontal component: x = v_0xt
• Vertical component: y = v_0yt - (1/2)gt²
• Position: r = xi + yj
• Horizontal velocity: v_x = v_0x
• Vertical velocity: v_y = v_0y - gt
• Velocity: v = v_xi + v_yj
• Time to reach maximum range: t = 2v_0y / g = 2v₀sin θ / g
• Maximum range: x_max = v₀²sin(2θ) / g
• Time to reach maximum height: t = v₀sin θ / g
• Maximum height: y_max = v₀²sin²θ / (2g)

Superposition of Uniform Motions

• Δr = Δxi + Δyj
• v = v_xi + v_yj
• x = v_xt = v_0xt

Uniform Circular Motion (MCU)

• ω = Δθ / Δt
• v = Δs / Δt
• Angular position: θ = θ₀ ± ωt
• Period: T = t / number of turns = 1 / frequency
• Frequency: f = number of turns / t = 1 / T
• ω = 2π / T = 2πf
• Centripetal acceleration: a_c = ω²r = (2π / T)²r

Uniformly Accelerated Circular Motion (MCUA)

• α = Δω / Δt
• Angular velocity: ω = ω₀ + αt
• Angular position: θ = θ₀ + ω₀t + (1/2)αt²