Linear Equations

Acceleration:- Acceleration of a body is defined as the rate of change of velocity.

Acceleration = change in velocity / time taken for change
                                     
            A= (v –u)/t

Where a = acceleration, v = final velocity, u = initial velocity, t = time taken
      s.i. unit of acceleration is m/sec². it is a vector quantity.
     

Note – when a body is moving with constant velocity then the acceleration is zero.

Uniform acceleration:- If the velocity changes at uniform rate the acceleration will uniform.

  Example: – the motion of freely falling body, the motion of bicycle going down the slop of the road and the motion of a ball rolling down an inclined plan.

Non-uniform acceleration:- A body has non-uniform acceleration if its velocity increases by unequal amounts in equal interval of time.

  Note- The velocity time graph  for a body having non-uniform acceleration is a curved line .

Retardation:- If the velocity of a body decreases the acceleration will be negative and called retardation.

Average velocity:- The average velocity is given by arithmetic mean of initial velocity and final velocity.

Average velocity = (initial velocity + final velocity) / 2
 
                                     V = (u+v) /2

Equation of uniformly accelerated motion:- There are three equation of motion of those bodies which travel with a uniform acceleration.

1. First equation of motion:- Consider a body having initial velocity ‘u’. suppose it is subjected to a uniform acceleration ‘a’ so that after time ‘t’ its final velocity becomes ‘v’.
     
         We know that

Acceleration = change in velocity/ time    = (final velocity – initial velocity)/ time taken
                              
   a =  ( v – u) /t
   at = v – u
    v = u + at
 
where   v = final velocity
u = initial velocity
a = acceleration
 t = time
                                    

2. second equation of motion:- suppose a body has an initial velocity ‘u’ and uniform acceleration ‘a’ for time ‘t’. let the distance traveled by the body in this time be ‘s’. then
        

average velocity = (initial velocity+ final velocity)/2
                                  =  (u + v) /2
We know that  
                            Distance = speed x time
                               S =  {(u + v) /2} X t

                       S = (u + u + at) X t /2   and get finally             

                  Where  s = distance traveled

u  = initial velocity
v =  final velocity
a = acceleration