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
s  distance (m, km, ft, miles)
t  time used (s, h)
Distance
v_{0}  initial velocity (m/s, ft/s)
a  acceleration (m/s^{2}, ft/s^{2})
t  time used (s, h)
Final Velocity
v_{0}  initial velocity (m/s, ft/s)
a  acceleration (m/s^{2}, ft/s^{2})
s  distance (m, ft)
Acceleration
v  final velocity (m/s, ft/s)
v_{0}  initial velocity (m/s, ft/s)
t  time used (s)
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})
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Dynamics
Motion  velocity and acceleration, forces and torque. 
Mechanics
Forces, acceleration, displacement, vectors, motion, momentum, energy of objects and more.
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