Derivation of equations of uniformly accelerated motion

1. First Equation of Motion

               The first equation of motion is :
 V = u + at. It gives the velocity acquired by a body in time t. We will now derive this 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' it's final velocity becomes 'v'. Now , from the definition of acceleration we know that : 
    
        Acceleration  = Change in velocity                                                      Time taken 

Or   Acceleration  =   final velocity- Initial velocity                                 Time taken 

    So,              a = v - u 
                                 t
                       at = v - u
   and,             v = u + at 
           Where     v = final velocity of the body 
                           u = Initial velocity of the body 
                             a = acceleration 
                   and     t = time taken 
         The equation v = u + at is known as the first equation of motion and it is used to find out the velocity 'v' acquired by a body in time 't', the body having an initial velocity 'u' and a uniform acceleration 'a'. In fact, this equation has four values in it, if any three values are known, the fourth value can be calculated. By  paying due attention to the sign of acceleration, this equation can also be applied to the problems of retardation. 
2. Second Equation of Motion 
           The second equation of motion is :  s = ut + 1/2 at the power 2. It gives the distance traveled by a body in time t. 
Let us derive this second equation of motion. 
        Suppose a body has an initial velocity 'u' and a uniform acceleration 'a' for time 't' so that its final velocity becomes 'v'. Let the  distance travelled by the body in this time be 's'. The  distance travelled by a moving body in time 't' can be found out by considering 
its average velocity. Since the initial velocity of the body is 'u' and it final velocity is 'v', the average velocity is given by : 
Average velocity  =  Initial velocity +Final velocity                                   2
        
That is,                    Average velocity  = u + v
                                                                    2
Also,         
Distance travelled = Average velocity ×Time
     So,      s  = ( u + v ) × t                        .....(1)
                              2
From the first equation of motion we have,
 v = u + at .  Putting the value of v in equation (1 ), we get :
                          s =  (u + u +at) × t
                                            2
                    s = (2u +at ) × t
                                     2
     s = 2ut + at the  power 2 
                          2
            s = ut + 1/2 at, the power 2
 Where      s = distance travelled by the body in time t 
          u = Initial velocity of the body 
           a = acceleration 
3. Third Equation of Motion
           The third equation of motion is : v to the power 2 = u the power 2 + 2as. It gives the velocity acquired by a body in travelling a distance s. We will now derive this third equation of motion.
         The third equation of motion can be obtained by eliminating t between the first equation of motion. This is done as follows.
           From the second equation of motion we have :
                   s = ut + 1/2at the power 2    ....(1)
And from the first equation of motion we have :                v = u + at                                 ....(2)
    This can be rearranged and written as :
                                       at = v -u 
                             Or   t = v - u 
                                             a
Putting this value of t in equation (1), we get : 
    s = u(v - u ) + 1/2a( v - u ) the whole power 2
                a                    (  a )
Or    s = uv - u the power 2 + a(v the power 2 +
                              a                      2a the power 2

u the power 2 - 2uv )        ( because ( v - u )the whole power 2 = v the whole power 2 + u the whole power 2 - 2vu )
 Or      s = uv - u the power 2 + v the power 2 +
                                 a                            2a 

u the power 2 - 2 uv

Or    s = 2uv - 2u the power 2 + v the power 2 +
                                               2a 
u the power 2 -2uv
 
Or                2as = v the power 2 - u the power 2
    Or         v the power 2 = u the power 2 + 2as

  Where           v = final velocity 
                         u = Initial velocity 
                           a = acceleration 
                  And  s = distance travelled 

                              

I am Saheb Sarkar from purnea .

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