Kinematics in One Dimension - Review

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kinematics - deals with the description of motion, without reference to the causes of the motion.

 

A.  Speed (scalar quantity) :

          the average speed, v, of an object is defined as the distance, d, it travels divided by the time, t, it takes to travel that distance

 

                    average speed = distance / time   or  v = d / t

 

Example:  What is the average speed of a car travelling at 450 km in 10.0 hours?

 

          v = d / t  = 450 km / 10.0 h  = 45 km/h

 

During the trip, the car’s speed would vary from minute to minute, its speed at a particular instant, as shown by the speedometer is called instantaneous speed, v.

 

B.  Velocity  (vector quantity)

          velocity is an objects is its speed in a particular direction, and is displacement (vector quantity), d, divided by time.  Since velocity is a vector, it can have a positive or negative value.  The sign signifies the direction of travel.

 

                    Average velocity = displacement / time   or v = d / t

 

Example :  If you travel 400.0 km south, then return 200.0 km north, and the trip takes 10.0 hours, what is your average velocity?

 

                    v = d / t = 400.0 - 200.0 km / 10.0 hours = 20 km/h

 

C.  Acceleration (vector quantity)

          a body accelerates when it changes speed or when it changes direction (or both).  Average acceleration is defined as the change in velocity divided by the time elapsed during the change in velocity.

 

average acceleration = change in velocity / time interval  

 

a = Dv / Dt = vf - vo / tf - to


Uniform Acceleration & The Four Equations Used -

Start with the velocity vs time graph of a uniformly accelerated object

 

slope = rise  =   vf - vo

  run         t - 0

 

slope = acceleration

 

a = vf - vo

t

equation for the straight line graph :  y = mx + b

 


          vf = at  + vo   or    vf = vo + at

 

 

the distance a uniformly accelerating body travels during t is :

 


          d = vave t                vave = vo + vf

  2

so

          d = vo + vf t  

                   2

 

since vf = vo + at

         

          d = vo + vo + at t          =    2vo + at t 

                     2                                   2

 

          d  = 2vot  + at2

                      2

 


          d = vot + ½at2

 

 

 

If time is not known, we can eliminated it by using equation 1 & 2

 

vf = vo + at

 

          solve for t

 

t = vf - vo              

          a

         

substitute into

 


d = vo + vf t        =   vo + vf    vf - vo

            2                        2           a

 

d = vf2 - vo2

          2a

 

          solving for vf2

 

 


                    vf2 = vo2 + 2ad