# Formulas of Motion - Linear and Circular

## Linear and angular acceleration, velocity and displacement

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### Linear Motion Formulas

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

where

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

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

Linear displacement can be expressed as (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}+ 2a 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)

#### 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 Formulas

#### 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 also be expressed as (angular acceleration = constant):

ω = ω_{o}+ α t (2b)

where

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

α = angular acceleration (rad/s^{2})

#### Angular Displacement

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

θ = ω_{o}t + 1/2 α t^{2}(2c)

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}(2d)where

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)where

T = angular momentum or torque (Nm)

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

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