y P~le's position at time Iz · ........P2,N ,t /11 ~ AT rvav = Ki I r. ,'.fl ....." PI P~cle'8 position /~etl I x Particle's palh 3.3 The vectors V, and Vz are the instanta· neous velocities at the points P, and Pz shown in Fig. 3.2. y L rPz t~3-~ The instantaneous \ velocity vector vis \ / tangent to Ihe pa;!, ~ at each point '--... v, o ~ PI /' ( -----------x Particle's palh 3.1 Position and Velocity Vedors To describe the motion of a particle in space, we must first be able to describe the particle's position. Consider a particle that is at a point P at a certain instant. The position vector 1 of the particle at this instant is a vector that goes from the ori-gin of the coordinate system to the point P (Fig. 3.1). The Cartesian coordinates x, y, and z of point P are the x-, y-, and z-components of vector 1. Using the unit vectors we introduced in Section 1.9, we can write 1=xi+yJ+zk (position vector) (3.1) During a time intervall1t the particle moves from PI, where its position vector is 110 to P2, where its position vector is '2• The change in position (the displace-ment) during this interval is 111 = 12 -11 = (X2 -xl)i + (Y2 -Yt)J + (Z2 -zl)i. We define the average velocity vav during this interval in the same way we did in Chapter 2 for straight-line motion, as the displacement divided by the time interval: (average velocity vector) (3.2) Dividing a vector by a scalar is really a special case of multiplying a vector by a scalar, described in Section 1.7; the average velocity Vav is equal to the dis-placement vector 111 multiplied by 1/l1t, the reciprocal of the time interval. Note that the x-component ofEq. (3.2) is Vav .• = (X2 -XI)/(t2 -tl) = I1x/l1t. This is just Eq. (2.2), the expression for average x-velocity that we found in Sec-tion 2.1 for one-dimensional motion. We now define instantaneous velocity just as we did in Chapter 2: It is the limit of the average velocity as the time interval approaches zero, and it equals the instantaneous rate of change of position with time. The key difference is that position 1 and instantaneous velocity V are now both vectors: 11-+ d1 v=lim r b.,-+O I1t dt (instantaneous velocity vector) (3.3) The magnitude of the vector V at any instant is the speed v of the particle at that instant. The direction of v at any instant is the same as the direction in which the particle is moving at that instant. Note that as I1t ~ 0, points PI and P2 in Fig. 3.2 move closer and closer together. In this limit, the vector 111 becomes tangent to the path. The direction of 111 in the limit is also the direction of the instantaneous velocity v. This leads to an important conclusion: At every point along the path, the instantaneous veloc-ity vector is tangent to the path at that point (Fig. 3.3). It's often easiest to calculate the instantaneous velocity vector using compo-nents. During any displacement 111, the changes 11x, l1y, and I1z in the three coordinates of the particle are the components of 111.