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Formulas of Motion

Linear and angular acceleration, velocity and displacement

car average velocity speed

Velocity can be expressed as (velocity is constant):

v = s / t (1a)

where

v = velocity (m/s, ft/s)

s = linear displacement (m, ft)

t = time (s)

Velocity can be expressed as (acceleration is constant):

v = v₀ + a t (1b)

where

v₀ = initial linear velocity (m/s, ft/s)

a = acceleration (m/s², ft/s²)

Linear displacement can be expressed as (acceleration is constant):

s = v₀ t + 1/2 a t² (1c)

Combining 1a and 1c to express the final velocity

v = (v₀² + 2 a s)^1/2 (1d)

Velocity can be expressed as (velocity is variable)

v = ds / dt (1f)

where

ds = change of displacement (m, ft)

dt = change in time (s)

Acceleration can be expressed as

a = dv / dt (1g)

where

dv = change in velocity (m/s, ft/s)

angular velocity

Angular velocity can be expressed as (angular velocity = constant):

ω = θ / t (2a)

where

ω= angular velocity (rad/s)

θ = angular displacement (rad)

t = time (s)

Angular velocity can be expressed as (angular acceleration is constant):

ω = ω_o + α t (2b)

where

ω_o = angular velocity at time zero (rad/s)

α = angular acceleration (rad/s²)

Angular displacement can be expressed as (angular acceleration is constant):

θ = ω_o t + 1/2 α t² (2c)

Combining 2a and 2c:

ω = (ω_o² + 2 α θ)^1/2

Angular acceleration can be expressed as:

α = dω / dt = d²θ / dt² (2d)

where

dθ = change of angular displacement (rad)

dt = change in time (s)