POSITION AND COORDINATES
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to specify a position, need:
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position along a line:
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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:
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position specified by one (signed) number
position in a plane:
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reference point (“origin”) O,
distance from origin
direction from origin
(to define direction, need reference direction(s)
need a third number (e.g. height above the x-y
plane)
coordinates:
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= set of numbers to describe position of a point
VECTORS AND SCALARS
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physical quantities can be “scalars”, “vectors”,
“tensors”, ......
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scalar:
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quantity for whose specification one number is sufficient;
examples: mass, charge, energy, temperature, volume,
density
vector:
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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
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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;
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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)
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 = 2  f (f = frequency of rotation)
ACCELERATION
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acceleration = rate of change of velocity
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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