Let a particle travel a distance 
as a function of time 
(here, 
can be thought of as the arc length
of the curve traced out by the particle). The speed (the
scalar norm of the vector velocity) is then given by
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(1)
|
The acceleration is defined as the time derivative of the velocity, so the scalar acceleration is given by
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(2)
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(3)
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(4)
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(5)
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(6)
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The vector acceleration is given by
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(7)
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(8)
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(9)
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where 
is the unit tangent
vector, 
the curvature, 
the arc length, and 
the unit normal
vector.
Let a particle move along a straight line so that the positions at times 
,
, and 
are 
, 
, and 
, respectively. Then the particle is uniformly accelerated
with acceleration 
iff
![a=2[((s_2-s_3)t_1+(s_3-s_1)t_2+(s_1-s_2)t_3)/((t_1-t_2)(t_2-t_3)(t_3-t_1))]](/images/equations/Acceleration/NumberedEquation2.svg) |
(10)
|
is a constant (Klamkin 1995, 1996).
Consider the measurement of acceleration in a rotating reference frame. Apply the rotation operator
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(11)
|
twice to the radius vector 
and suppress the body notation,
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(12)
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(13)
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(14)
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(15)
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(16)
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Grouping terms and using the definitions of the velocity 
and angular
velocity 
gives the expression
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(17)
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Now, we can identify the expression as consisting of three terms
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(18)
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(19)
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(20)
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a "body" acceleration, centrifugal acceleration, and Coriolis acceleration. Using these definitions finally gives
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(21)
|
where the fourth term will vanish in a uniformly rotating frame of reference (i.e., 
). The centrifugal acceleration
is familiar to riders of merry-go-rounds, and the Coriolis acceleration is responsible
for the motions of hurricanes on Earth and necessitates large trajectory corrections
for intercontinental ballistic missiles.
