POSITION AND COORDINATES

to specify a position, need: reference point (“origin”) O, distance from origin direction from origin (to

define direction, need reference direction(s) position along a line:

position specified by one (signed) number position in a plane:

position of point P specified by length of “vector” OP (distance)and angle of OP with respect to reference direction,

or by two numbers x,y position in 3-dimensional space:

need a third number (e.g. height above the x-y plane)

coordinates: = set of numbers to describe position of a point

VECTORS AND SCALARS physical quantities can be “scalars”, “vectors”, “tensors”, ...... scalar:

quantity for whose specification one number is sufficient;

examples: mass, charge, energy, temperature, volume, density

vector: quantity for whose specification one needs:

magnitude (one number) direction (number of numbers depends on

dimension) numbers specifying vector: “components of the

vector” in suitably chosen coordinate system; e.g. components of the position vector: numbers

specifying the position; examples:

position vector, velocity, acceleration, momentum, force, electric field,..

magnitude = “length of vector”e.g.

distance from reference point” = magnitude of position vector,

“speed” = magnitude of velocity.

velocity

velocity: = (change in position)/(time interval) average velocity = velocity evaluated over finite

(possibly long) time interval vav = x/t, x = total distance travelled during time interval t (including speeding up, slowing down, stops,...);

instantaneous velocity = velocity measured over very short time interval ;

ideally, t = 0, i.e. time interval of zero length: v = limit of (x/t) for t 0;

t 0 is limit of t becoming “infinitesimally small”, “t approaches zero”, “t goes to zero”;

note that velocity is really a vector quantity (have considered motion in only one dimension)

difference quotient: x/t = “difference quotient”

of position with respect to time difference quotient = ratio of two differences; limit for t 0:

[limit of (x/t) for t 0] = dt/dx = “differential quotient”,

also called “derivative of x with respect to t” “differential calculus” = branch of mathematics,

about how to calculate differential quotients. angular velocity : (change in angle)/(time

interval) = 2 f (f = frequency of rotation)

ACCELERATION

acceleration = rate of change of velocity a = (change in velocity)/time interval average acceleration aav= v/t ,

v = change in velocity t = duration of time interval

for this change instantaneous acceleration

= limit of average acceleration for infinitesimally short time interval ,

a = dv/dt acceleration, like velocity, is really a vector

quantity change of velocity without change of speed:

if only direction changes, with speed staying the same;

e.g. circular motion if a = 0: no acceleration,

velocity constant “uniform motion” motion in straight line with constant

speed angular acceleration = rate of change of

angular velocity