IF A PARTICLE MOVES IN A STRAIGHT LINE WITH CONSTANT ACCLERATION DESCRIBES DISTANCE S1 AND S2 IN 2  SUCESIVE INTERVALS OF TIME T1 AND T2; PROVE THAT ACCLERATION IS

   2(T1S2 - T2S1)      OR 2(T1S2 - T2S1) / T1T2(T1+T2)

     T1T2(T1+T2)

 a = (u2-u1)/T1 = (u3 - u2)/T2 .........................eq(1)

 

  S1 = u1T1 + 1/2. a. T1²

       = u1T1 + 1/2 . (u2-u1)/T1 . T1²

       = u1T1 - 1/2 . u1T1  + 1/2.u2T1

       = 1/2. (u1+u2)T1

 

 So,   u1+u2 = 2S1/T1

 Also, u2 - u1 = aT1  from eq(1)

 

  Thus u2 + u1 + u2 - u1 = aT1 + 2S1/T1

            2u2 = at1 +  2S1/T1

              u2 = aT1/2 + S1/T1

 

 Now,

            S2 = u2 + 1/2. a. T2²

            S2 = (aT1/2 + S1/T1).T2 + 1/2.aT2²

            S2 - S1.T2/T1 = aT1.T2/2 + a.T2²/2

   So,  a = 2{S2T1 - S1T2}/(T1²T2 + T2²T1)

             = 2(T1S2 - T2S1) / T1T2(T1+T2)