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∆r
v avg = t ∆ 2-1

The instantaneous velocity is determined from this equation by
letting t∆→ 0, and consequently the direction of ∆r approaches the
tangent to the curve. Hence,

∆r
v = lim v = lim 2-2
t ∆→ 0 avg t ∆→ 0 t ∆
or
d r
v = =r ɺ 2-3
dt
Since dr will be tangent to the curve, the direction of v is also
tangent to the path, Fig. 2-2,c. The magnitude of v is called the speed.
Thus, the speed can be obtained by differentiating the position
vector with respect to time. The “dot” notation r ɺ represents the first
time derivatives.

a b c
Fig. 2-2

Acceleration. If the particle has a velocity v at time t and a
velocity ′ v = v+ ∆v at t + t ∆ Fig. 2-3,a, then the average acceleration
of the particle during the time interval t∆ is

∆v
a avg = t ∆ 2-4

′
where ∆v = v -v. To study this time rate of change, the two velocity
vectors in Fig. 2-3,a are plotted in Fig. 2-3,b such that their tails are
located at the fixed point O′ and their arrowheads touch points on a
curve. This curve is called a hodograph, and when constructed, it
describes the locus of points for the arrowhead of the velocity vector
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