The velocity vector is in the direction of the tangent to the curve and the com-
ponent of the velocity along the direction 0 P is called the radial component of the
velocity and denoted vr. At the point P construct a line perpendicular to the line
segment 0 P, then the component of the velocity projected onto this perpendicular
line segment is called the transverse component of the velocity and denoted by vθ.
These projections of the velocity vector give the radial and transverse components
vr=vcos ψ and vθ=vsin ψ
where v=
ds
dt =
√
v^2 r+v^2 θ is the magnitude of the velocity called the speed of the
particle. The unit tangent vector to the curve is given by
ˆet= cos ψˆer+ sin ψˆeθ
where ˆer is a unit vector in the radial direction and ˆeθ is a unit vector in the
transverse direction. The derivative of the position vector with respect to the time
tcan be written as
dr
dt =
dr
ds
ds
dt =vˆet=vcos ψˆer+vsin ψˆeθ=vrˆer+vθeˆθ
Therefore, when the position vector to the point P is written in the form
r =xˆe 1 +yˆe 2 or r =rcos θˆe 1 +rsin θˆe 2 =reˆr
where ˆeris the unit vector in the radial direction given by ˆer= cos θˆe 1 + sin θˆe 2 , one
finds the derivative of this unit vector with respect to θproduces the vector
dˆer
dθ =
ˆeθ=−sin θeˆ 1 + cos θˆe 2
The derivative of the position vector with respect to arc length is a unit vector so
that
dr
ds
=dx
ds
eˆ 1 +dy
ds
ˆe 2 =rdˆer
ds
+dr
ds
ˆer=eˆt= cos ψˆer+ sin ψeˆθ
where dˆer
ds
=−sin θdθ
ds
ˆe 1 + cos θdθ
ds
eˆ 2 =ˆeθdθ
ds
Therefore,
dr
ds
=r
dθ
ds
ˆeθ+
dr
ds
ˆer=ˆet= cos ψˆer+ sin ψeˆθ
