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0593_C03_fm Page 61 Monday, May 6, 2002 2:03 PM

Kinematics of a Particle 61

coordinates of P relative to X, Y, and Z. Then, the position velocity and acceleration of P
may be expressed in terms of N , N , and N as:
Y
X
Z
p = x N + y N + z N
X Y Z
V = x ˙ N + y ˙ N + z ˙ N
P
X y Z
and

p
a = ˙˙ N + ˙˙ N + ˙˙ z N (3.3.5)
x
y
X y Z

3.4 Relative Velocity and Relative Acceleration

Consider again the movement of a particle P represented by a point P in a reference frame
R. As before, let the position vector p locate P relative to a fixed point O in R as in Figure
3.4.1. From Eq. (3.3.1), the derivative of p in R is the velocity of P in R. That is,

V = d p dt (3.4.1)
P
where we have deleted the superscript R in dp/dt because the reference frame is clearly
understood. Because O is fixed in R, V is often called the absolute velocity of P in R.
P
Consider now a second particle Q represented by point Q also moving in R. Let q locate
Q relative to O as in Figure 3.4.2. Then, as with P, the velocity of Q in R is:

V = d q dt (3.4.2)
Q
Again, because O is fixed in R, V is often called the absolute velocity of Q in R.
Q
The difference in the absolute velocities of P and Q is called the relative velocity of P with
respect to Q in R and is written as V P/Q or as V P/Q . That is,
P
P
/
V PQ = V − V Q (3.4.3)
Q
r
P q
P
R p R p
O O

FIGURE 3.4.1 FIGURE 3.4.2
A point P moving in a reference frame R. Points P and Q moving in reference frame R.