Derive the expression for displacement in the nth second.

# Derive the expression for displacement in the nth second

a = accelleration

v = velocity, u=initial velocity

s = distance

s = ut + 1/2at^{2}

To calculate the distance traveled during the nth second, you calculate the distance covered in n seconds and subtract the distance covered in n-1 seconds and get

s = un-u(n-1) + 1/2an^{2}-1/2a(n-1)^{2}

simplifying gives us

u term = u(n-(n-1)) = u*1 = u

(n-1)^{2} = n^{2}-2n+1

a term = 1/2a(n^{2}-(n-1)^{2}) = 1/2a(n^{2}-n^{2}+2n-1) the n^{2}’s cancel and give us 1/2a(2n-1) .

Final equation for displacement in the nth second is

s = u + 1/2a(2n-1)