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

• average velocity
  = velocity evaluated over finite (possibly long) time interval;

  

  = total distance travelled during time interval
  (including speeding up, slowing down, stops,...);

• instantaneous velocity
  = velocity measured over very short time interval ;
  ideally, , i.e. time interval of zero length:

  

  : "limit of becoming infinitesimally small'',
  " approaches zero'', `` goes to zero'',

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

  

  of position with respect to time;
  difference quotient = ratio of two differences;

• limit for :

  

  also called ``derivative of x with respect to t''

• ``differential calculus'' = branch of mathematics,
  about how to calculate differential quotients

ACCELERATION

• acceleration = rate of change of velocity

  

• average acceleration

  

  change in velocity
  duration of time interval for this change

• instantaneous acceleration
  = limit of average acceleration for infinitesimally short time interval
• acceleration, like velocity, is really a
  vector quantity
• can have change of velocity without change of speed:
  if only direction changes, with speed staying the same;
  e.g. circular motion
• if : no acceleration,
  velocity constant
  ``uniform motion''
  

  motion in straight line with constant speed

UNIFORM MOTION

• uniform motion = motion with constant velocity

  

  i.e. motion with constant speed and constant direction;
  constant direction motion is in straight line;

  

  = present time - start time =
  = present position - start position =
  displacement
  

• distance travelled:
  distance travelled = magnitude of displacement vector;
  for simplicity, assume
  

  

  where t = time since start = duration of trip

• graph of velocity vs time: straight horizontal line (zero slope)
• graph of distance vs time: straight line, slope = speed v
• = area under line of

MOTION WITH CONSTANT ACCELERATION

change in velocity = acceleration times elapsed time;

• for simplicity, assume body originally at rest,
  i.e. initial velocity , and

  

  = velocity at time t, where t = time since acceleration began;

• now consider only motion in one dimension - no vector signs;
• graph of acceleration vs time: straight horizontal line (zero slope)
• graph of velocity vs time: straight line with slope a
• graph of distance vs time:
  "curved line", with changing slope;
  slope at every moment (point in time) = speed at that moment;