# Formulas of Motion - Linear and Circular

### 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}) *

### Linear Motion Calculators

#### Average velocity

#### Distance

#### Final Velocity

#### Acceleration

### 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

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 = α I (2f)where

T = angular moment or torque (N m)

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

## Related Topics

### • Dynamics

Motion - velocity and acceleration, forces and torque.

### • Mechanics

Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.

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