Question

Derive the equation.Derive the equation for uniform acceleration motion for the displacement covered in nth second of its motion.(sn=u a(n-1/2)

Answer 1

From the equations of kinematics we have that :  s= u t + 1/2 a t²

  Displacement during the first n sec,     = S_n = u * n + 1/2 * a * n²
  Displacement during the first n-1 sec.  = S_n-1 = u * (n-1) + 1/2 * a * (n-1)²
 
  displacement during n th sec. = S_n - S_n-1
       = u n + 1/2 a n² - u n + u - 1/2 a n² + 1/2 a * 2n - 1/2 a * 1
       = u + a ( n - 1/2)

that is the answer.

Answer 2

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$\mathfrak{\huge{\underline{Answer:-}}}$

→ Displacement in nth second S_n = u + a(n - 1/2).

$\huge{\mathscr{\boxed{\boxed{Explanation:-}}}}$

$\rm{\underline{\boxed{Derivation:}}}$

=> Let ‘S_n’ be the displacement of a body in “n” seconds. i.e., t = n

=> Similarly, Let ‘S_n-1’ be the displacement if the body in (n - 1) seconds. i.e., t = n - 1

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$\sf{We\: Know\:That,\:S\:=\:ut\:+\:1/2\:at^2\:(Kinematic\: Equation)}$

→ Then, we can write S_n = un+1/2 an² (equation 1)

→ And S_n-1 = u(n - 1) +1/2 a (n - 1)² (equation 2)

» From equation (1) & (2)

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S_n - S_n-1 = (un + 1/2 an²) - [u(n - 1) + 1/2 a (n - 1)²]

= (un + 1/2 an²) - [u(n - 1) + 1/2 a (n² + 1 - 2n)]

= (un + 1/2 an²) - [un - u + 1/2 an² + 1/2 a - an]

= un + 1/2 an² - un + u - 1/2 an² - 1/2 a + an

= u + an - 1/2 a

$\rm{S_n\:-\: S_n-1 \:=\:u\:+\:a\:(n\:-\:1/2)}$

➡️ Displacement in nth second S_n = u + a(n - 1/2)

$\huge{\bold{\boxed{\boxed{Hope\:It\:Helps\:You}}}}$