Concepts Key for the Lecture of 27Jun03 The state of a single-particle system is given by its position. The position can change with time. Position is expressed by the position vector (or radius vector) r(t). The position vector can be thought of as being an ordered triple of three coordinates, each a function of time r(t)
[ x(t), y(t), z(t) ] ! or as the sum of three vector components r ( t )
x ( t ) i x ( t ) j z ( t ) k ! In either approach, the position as a function of time is given by a set of three equations parametric in time, one for each coordinate: x(t), y(t), and z(t). In many problems, a clever choice of coordinate systems can reduce the number of coordinates to two or one, thus simplifying the calculations. Always look for the possibility of making this simplification. In one dimension you can drop the vector notation entirely and give the position simply by x(t) (or by y(t) or by z(t), or whatever coordinate you have chosen.) The velocity is the rate of change of position in space. Just as with speed, v
! lim ! r ! lim r ( t t ) r ( t ) ! ! d! r !
t0
t t0 t dt
Since velocity is a vector, it has three components v ( t )
vx ( t ) i vy ( t ) j vz ( t ) k !
Each of these velocity components is just the rate of change of the corresponding coordinate: v(t)
! dr(t) ! dx dy dz
i j k dt dt dt dt
Again, many problems can be reduced to two, or even one, dimension by a clever choice of coordinate systems. In one dimension you can drop the vector notation entirely. The velocity along one axis is just v(t) = dx/dt (or dy/dt or dz/dt, or whatever coordinate chosen).
Given velocity, the position is computed by integrating the velocity. For the x-coordinate, x(t)
v ( t )...