Formulas of Motion - Linear and Circular

Linear and angular (rotation) acceleration, velocity, speed and distance

Linear Motion Formulas

car average velocity speed

Average velocity/speed can be expressed as:

v = s / t         (1a)


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

s = linear distance traveled (m, ft)

t = time (s)

  • distance is the length of the path a body follows in moving from one point to another - displacement is the straight line distance between the initial and final positions of the body
  • we use velocity and speed interchangeable - but be aware that speed is a measure of how fast or slow a distance is covered, the rate at which distance is covered - velocity is a vector, specifying how fast or slow a distance is covered and the direction

Velocity can be expressed as (if acceleration is constant):

v = v0 + a t         (1b)


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

a = acceleration (m/s2, ft/s2)

Linear distance can be expressed as (if 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 distance (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 - Marathon

If a marathon - 42195 m - is run in 2:03:23 (7403 s) - the average speed can be calculated as

  v = (42195 m) / (7403 s)

     = 5.7 m/s

     = 20.5 km/h

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 - Rotation - Formulas

angular velocity

Angular Velocity

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

ω = θ / t         (2)


ω= angular velocity (rad/s)

θ = angular distance (rad)

t = time (s)

The tangential velocity of a point in angular velocity - in metric or imperial units like m/s or ft/s - can be calculated as

v = ω r    (2a)


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

r = distance from center to the point (m, ft, in)

Example - Tangential Velocity of a Tire on a Bicycle

A 26 inches bicycle wheel rotates with an angular velocity of π radians/s (0.5 turn per second). The tangential velocity of the tire can be calculated as

v = (π radians/s) ((26 in) / 2)

  = 40.8 in/s

Angular Velocity and Acceleration

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 distance 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 distance (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 (N m)

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

Example - Flywheel

flywheelBy Geni (Photo by User:geni) [GFDL or CC-BY-SA-3.0-2.5-2.0-1.0], via Wikimedia Commons

A flywheel is slowed down from 2000 rpm (revolutions/min) to 1800 rpm in 10 s. The deceleration of the flywheel can be calculated as

α = ((2000 revolutions/min)  - (1800 revolutions/min)) (0.01667 min/s) (2 π rad/revolution) / (10 s)

  = 2.1 rad/s2

Related Topics

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

Related Documents

Key Words

  • en: motion speed distance acceleration
  • es: movimiento de aceleraciĆ³n distancia Velocidad
  • de: Bewegungsgeschwindigkeit Abstand Beschleunigung

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