Question

Derive:
$s_{n} = u + \dfrac{1}{2}a(2n - 1)$
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Answer 1

So, S(nth) = displacement of body in nth second

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionally

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHS

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]2n-1= n=time = [T]

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]2n-1= n=time = [T]1/2=dimensionless = [M°L°T°]

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]2n-1= n=time = [T]1/2=dimensionless = [M°L°T°]----------------------------

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]2n-1= n=time = [T]1/2=dimensionless = [M°L°T°]----------------------------U+(1/2)*a*(2n-1)= [LT^-1]+[LT^-2]*[T]

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]2n-1= n=time = [T]1/2=dimensionless = [M°L°T°]----------------------------U+(1/2)*a*(2n-1)= [LT^-1]+[LT^-2]*[T][LT^-1]+[LT-^1]=[LT^-1] by principle of homogeneity.

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]2n-1= n=time = [T]1/2=dimensionless = [M°L°T°]----------------------------U+(1/2)*a*(2n-1)= [LT^-1]+[LT^-2]*[T][LT^-1]+[LT-^1]=[LT^-1] by principle of homogeneity.As LHS= RHS dimensionally i.e [LT^-1]

So, S(nth) = displacement of body in nth secondi.e S(nth) = [L]/[T] = [LT^-1] dimensionallyRHSNow, U+(1/2)*a*(2n-1) = where,U= initial velocity = [LT^-1]a= acceleration = [LT^-2]2n-1= n=time = [T]1/2=dimensionless = [M°L°T°]----------------------------U+(1/2)*a*(2n-1)= [LT^-1]+[LT^-2]*[T][LT^-1]+[LT-^1]=[LT^-1] by principle of homogeneity.As LHS= RHS dimensionally i.e [LT^-1]If it is wrong then Sorry!