Formulas of Motion - Linear and Circular

Linear and angular acceleration, velocity and displacement

Linear Motion Formulas

car average velocity speed

Velocity can be expressed as (velocity is constant):

v = s / t         (1a)


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)


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 1b 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)


ds = change of displacement (m, ft)

dt = change in time (s)

Acceleration can be expressed as

a = dv / dt         (1g)


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

Example - Acceleration of a Car

A car accelerates from 0 km/h to 100 km/h in 10 seconds. The acceleration can be calculated by transforming 1b to

a = (v - v0) / t

   = ( (100 km/h) (1000 m/km) / (3600 s/h) - (0 km/h) (1000 m/km) / (3600 s/h) ) / (10 s)

   = 2.78 (m/s2)

Circular Motion Formulas

angular velocity

Angular Velocity

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

ω = θ / t         (2a)


ω= angular velocity (rad/s)

θ = angular displacement (rad)

t = time (s)

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

ω = ωo + α t         (2b)


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

α = angular acceleration (rad/s2)

Angular Displacement

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

θ = ωo t + 1/2 α t2         (2c)

Combining 2a and 2c:

ω = (ωo2 + 2 α θ)1/2

Angular Acceleration

Angular acceleration can be expressed as:

α = dω / dt = d2θ / dt2         (2d)


dθ = change of angular displacement (rad)

dt = change in time (s)

Angular Momentum or Torque

Angular momentum or torque can be expressed as:

T = ω I          (2e)


T = angular momentum or torque (Nm)

I = moment of inertia (lbm ft2, kg m2)


Related Topics

  • Dynamics - Dynamics Motion - velocity and acceleration
  • Mechanics - Kinematics, forces, vectors, motion, momentum, energy and the dynamics of objects

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