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0593_C03_fm Page 59 Monday, May 6, 2002 2:03 PM

Kinematics of a Particle 59

Observe that when t is zero, p and dp/dt are:

p = 0 and d p dt = −2 n − 3 n + 4 n m sec (t = 0) (3.2.9)
1 2 3

Alternatively, when t is one, p and dp/dt are:

p =− n − 2 n + 3 n m and d p dt = 0 = (t 1 sec ) (3.2.10)
2
1
3
This shows that a vector may be zero while its derivative is not zero, and conversely the
derivative may be zero while the vector is not zero.
As noted in Eq. (3.2.5), vector derivatives may be expressed as linear combinations of
scalar derivatives. This means that the algebra associated with scalar derivatives is appli-
cable with vector derivatives as well. For example, the rules for differentiating sums and
products are the same for scalars and vectors. Also, the chain rule is the same. Specifically,
let V and W be vector functions of t, and let s be a scalar function of t. Then from Eq.
(3.2.5) it is readily seen that:

(
+
+
d VW)/ dt = dV dt dW dt (3.2.11)
( dt) = ( ds dt V ) + ( dt)
d sV s dV (3.2.12)

dt W V⋅(
(
⋅
d VW)/ dt = ( dV ) ⋅ + dW dt) (3.2.13)
and
( × dt = dt) × W V ×( dt)
+
d VW dV dW (3.2.14)

Finally, if V is a function of s and if s is a function of t, we have:
ds ds dt)
dV dt = ( dV )( (3.2.15)

3.3 Position, Velocity, and Acceleration
Consider the movement of a particle P along a curve C as in Figure 3.3.1. Let P be a point
representing particle P. Let p locate P relative to a fixed point O of a reference frame R as
P
shown. The position of P is designated by the vector p. The velocity of P, V , is defined as
the time derivative of p in R. That is,

V = R d p dt (3.3.1)
P

P
Similarly, the acceleration of P is defined as the time derivative of V in R. That is,
a = R d V dt = R d 2 p dt 2 (3.3.2)
P
P