# 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 = v_{0}+ a t (1b)

where

v_{0}= initial linear velocity (m/s, ft/s)

a = acceleration (m/s^{2}, ft/s^{2})

Linear distance can be expressed as (if acceleration is constant):

s = v_{0}t + 1/2 a t^{2}(1c)

Combining *1b* and *1c* to express the final velocity

v = (v_{0}^{2}+ 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 - v _{0}) / t*

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

* = 2.78 (m/s ^{2})*

### 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 = ω_{o}+ α t (2c)

where

ω_{o}= angular velocity at time zero (rad/s)

α = angular acceleration or deceleration (rad/s^{2})

#### Angular Displacement

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

θ = ω_{o}t + 1/2 α t^{2}(2d)

Combining 2a and 2c:

ω = (ω_{o}^{2}+ 2 α θ)^{1/2}

#### Angular Acceleration

Angular acceleration can be expressed as:

α = dω / dt = d^{2}θ / dt^{2}(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/s^{2}*

* = (2.1 rad/s ^{2}) (360 / (2 π) degrees/rad) *

* = 120 degrees/s ^{2}*

#### Angular Moment - or Torque

Angular moment or torque can be expressed as:

T =αI (2f)where

T = angular moment or torque (N m)

I= Moment of inertia (lb_{m}ft^{2}, kg m^{2})

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