## Time Derivatives of Vectors

In many instances in movement and motion, velocity of a point in the body or even the angular velocity of a body segment may be a function of time. If either the magnitude or the direction of a vector v depends on time t, then the vector is said to be a function of t. The time derivative of vectors depends on the coordinate system in which they are measured. Let P be a vector and let two Cartesian reference frames, E and B, be defined by their respective unit vectors (e1, e2, e3) and (b1, b2, b3). Then P can be expressed using the unit vectors of either E or B:

where EPi and BPi represent the projections of the vector P along the unit vectors of the coordinate systems E and B, respectively. The time derivative of P in E (and in B) is defined as

EdP/dt = {dEPx/dt) e1 + {dEP2/dt) e2 + (dEP3/dt) e3 (9.2a)

BdP/dt = (d%/dt) b1 + (dBP2/dt) b2 + (d%/dt) b3 (9.2b)

As an example to illustrate the use of these equations, consider an aerobic instructor asking the class to extend their arms to the side and then rotate them around a horizontal axis that connects the two shoulder joints (Fig. 9.1). The motion involved can be represented by a slender rod rotating around a fixed axis. The position vector rH/S from the shoulder joint to the hand can be written in the reference frame B as follows:

where L denotes the length of the upper arm. The same position vector is a function of time in the inertial reference frame E:

in which the angles 6 and \$ have been defined in the figure. If the arm rotates such that the angle 6 remains constant, the time derivative of P in B and in E can be written as

Clearly in the inertial reference frame E the line segment connecting the shoulder to the hand changes orientation with time and this reflects in the time derivative in the reference frame E. Figure 9.1. As part of the warm-up routine, an aerobic instructor rotates her arms around a horizontal axis that passes through her shoulder joints. The inertial coordinate system E is fixed on her trunk whereas the Cartesian coordinates B moves with the rotating arms. The unit vector along the arm length is denoted as b1.

Figure 9.1. As part of the warm-up routine, an aerobic instructor rotates her arms around a horizontal axis that passes through her shoulder joints. The inertial coordinate system E is fixed on her trunk whereas the Cartesian coordinates B moves with the rotating arms. The unit vector along the arm length is denoted as b1. 