Derivation of Normal-Tangential Velocity and Acceleration for Curvilinear
Motion
We set up the coordinate system with the tangential unit direction tangent to the direction of motion
and the normal direction point towards the center of the curve, Figure 1.
Figure 1. Normal and tangential coordinate system
The particle moves from an initial position to a new position, as shown in Figure 2.
Figure 2. Displacement of a particle in normal-tangential coordinates
The velocity is the change in position over time, so the velocity is (by definition) tangent to the curve:
dr ds
 uˆ t
dt dt
 v  vuˆ t
v
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Figure 3. Velocity tangent to path
Acceleration is the time rate of change of velocity. Using the product rule, we get:
dv d
 vuˆ t
dt dt
 a  vuˆ t  vuˆ t
a
v is the rate at which the speed is increasing or decreasing, or the tangential acceleration:
v  at
We also need uˆ t . We can determine the differential change in the tangential by looking at the geometry
of the curve. Remember, we are approximating the arc length ds by a circle. From Figure 4 (c) and
vector addition we can see the new normal tangential direction is:
ut  uˆ t  duˆ t
(a)
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(b)
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(c)
Figure 4. Schematic of normal and tangential coordinate system: change in tangential coordinate (a); highlight of region of
interest (b); close-up of region of interest (c)
We approximate the differential change duˆ t by an arc. The length of an arc is s  r , but in our case
the radius is uˆ t  1 . Note the direction of change is the normal direction uˆ n .
 duˆ t  1d uˆ n
We want the differential change with respect to time, therefore we have:
uˆ t   uˆ n
Finally, we have:
a  vuˆ t  v uˆ n
In normal-tangential coordinates, we typically have a path and a velocity, so we want to get ride of  .
We are approximating the curve by a arc:
ds   d

ds
d

dt
dt
v  
Or,
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
v

So acceleration in normal and tangential coordinates is,
a  at ut  anun  vut 
v2

un
an is the centripetal acceleration and at is the tangential acceleration (or increase/decrease in speed).
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