Acceleration

(d) Displacement in nth second of motion. Consider that an object starts moving with initial velocity u and the uniform acceleration a. Let the displacement of the object in the nth second of its motion be \( S_{nth} \)
Then ,     \( S_{nth} = S - S_{n - 1} \) .............(v)
where  S_n and S_n-1 are displacements of the object in n and n - 1 seconds. The values of S, and  can be obtained by putting t = n  in the equation 
        \( S = ut + \frac{1} {2}at^2 \).............(vi)
then  =un+\( \frac{1} {2} \)an²  ;
 For t=n-1 from (vi)
        S_n-1=u(n-1)+\( \frac{1} {2} \)a(n-1)²
Therefore, the equation (v) becomes
  \( \begin{gathered} S_{nth} = \left[ {un + \frac{1} {2}an^2 } \right] - \left[ {u\left( {n - 1} \right) + \frac{1} {2}a\left( {n - 1} \right)^2 } \right] \hfill \\ = \left[ {un + \frac{1} {2}an^2 } \right] - \left[ {u\left( {n - 1} \right) + \frac{1} {2}a\left( {n^2 + 1 - 2n} \right)} \right] \hfill \\ = u\left( {n - n + 2} \right)\frac{1} {2}a\left( {n^2 - n^2 - 1 + 2n} \right) \hfill \\ S_{nth} = u + \frac{1} {2}a\left( {2n - 1} \right) \hfill \\ \end{gathered} \)