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Formulas of Motion - Linear and Circular

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

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Linear Motion Formulas

Average velocity/speed of a moving object can be calculated as

v = s / t                            (1a)

where

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

If acceleration is constant then velocity can be expressed as:

v = v0 + a t                            (1b)

where

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)

where

ds = change in distance (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)

Example - a Marathon Run

If a marathon - 42195 m - is run in amazing 2:03:23 (7403 seconds) (Wilson Kipsang, Kenya - September 29, 2013 Berlin Marathon) - the average speed can be calculated

  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)

Linear Motion Calculators

Average velocity

s - distance (m, km, ft, miles)

t - time used (s, h)


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Distance

v0 - initial velocity (m/s, ft/s)

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

t - time used (s, h)


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Final Velocity

v0 - initial velocity (m/s, ft/s)

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

s - distance (m, ft)


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Acceleration

v - final velocity (m/s, ft/s)

v0 - initial velocity (m/s, ft/s)

t - time used (s)


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Circular Motion - Rotation

Angular Velocity

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

ω = θ / t                           (2)

where

ω = angular velocity (rad/s)

θ = angular distance (rad)

t = time (s)

Angular velocity and rpm:

ω = 2 π n / 60                        (2a)   

where

n = revolutions per minute (rpm)

π = 3.14...

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                              (2b)

where

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

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

Example - Tangential Velocity of a Bicycle Tire

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 inches) / 2)

  = 40.8 inches/s

Angular Velocity and Acceleration

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

ω = ωo + α t                               (2c)

where

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

α = angular acceleration or deceleration (rad/s2)

Angular Displacement

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

θ = ωo t + 1/2 α t2                                     (2d)

Combining 2a and 2c:

ω = (ωo2 + 2 α θ)1/2

Angular Acceleration

Angular acceleration can be expressed as:

α = dω / dt = d2θ / dt2                                    (2e)

where

dθ = change of angular distance (rad)

dt = change in time (s)

Example - Flywheel Deceleration

By 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 rev/min)  - (1800 rev/min)) (0.01667 min/s) (2 π rad/rev) / (10 s)

  = 2.1 rad/s2

  = (2.1 rad/s2) (360 / (2 π) degrees/rad) 

  = 120 degrees/s2

Angular Moment - or Torque

Angular moment or torque can be expressed as:

T = α I                                 (2f)

where

T = angular moment or torque (N m)

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

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