Formulas of Motion  Linear and Circular
Linear and angular (rotation) acceleration, velocity, speed and distance.
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
By Geni (Photo by User:geni) [ GFDL or CCBYSA3.02.52.01.0 ], 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 = 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.
Related Documents

Acceleration
Change in velocity vs. time used. 
Acceleration Units Converter
Converting between units of acceleration. 
Angular Motion  Power and Torque
Angular velocity and acceleration vs. power and torque. 
Banked Turn
A turn or change of direction in which the vehicle banks or inclines, usually towards the inside of the turn. 
Car  Required Power and Torque
Power, torque, efficiency and wheel force acting on a car. 
Car Acceleration
Car acceleration calculator. 
Centripetal and Centrifugal Acceleration Force
Forces due to circular motion and centripetal / centrifugal acceleration. 
ConnRod Mechanism
The connecting rod mechanism. 
Conservation of Momentum
The momentum of a body is the product of its mass and velocity  recoil calculator. 
Disk Brakes  Torque and Force
Forces and torque activated with disk brakes. 
Distance Traveled vs. Speed and Time  Calculator and Chart
Velocity plotted in time used diagram. 
Elevators  Force and Power
Required force and power to lift an elevator. 
Flywheels  Kinetic Energy
The kinetic energy stored in flywheels  the moment of inertia. 
Impulse and Impulse Force
Forces acting a very short time are called impulse forces. 
Kinetic Energy
Energy possessed by an object's motion is kinetic energy. 
Piston Engines  Displacement
Calculate piston engine displacement. 
Projectile Range
Calculate the range of a projectile  a motion in two dimensions. 
Radians vs. Degrees
Radian is the SI unit of angle. Convert between degrees and radians. Calculate angular velocity. 
Rolling Resistance
Rolling friction and rolling resistance. 
Vehicle  Distance Traveled vs. Velcocity and Time Used (mph)
Speed (mph) and time (hours) and distance traveled (miles) chart. 
Vehicle  Distance Traveled vs. Velocity and Time (km/h)
Speed (km/h) vs. time (hours) and distance traveled (km). 
Velocity Units Converter
Convert between speed and velocity units like m/s, km/h, knots, mph and ft/s.