Dimensional analysis is a procedure whereby fundamentals quantities in nature, and their units, called Fundamental units, are used to analyse the relationship between variables in a formula or equation to determine its accuracy.
The fundamental quantities and their units are:
Mass, M (kilogram, kg)
Length, L (metre, m)
Time, T (seconds, s)
For the equations to be dimensionally correct, all the variables in the equation must have the same dimension.
a) ½mv² = ½mv² + m²gh
v represents velocity, g represents accelerator due to gravity, h represents height.
M * (L / T)² = M * (L / T)² + M² * (L / T²) * L
ML² / T² = ML² / T² + M²L² / T²
The dimensions don't all match, hence, this equation cannot be correct.
b) v = u + at²
u represents velocity and a represents acceleration
L/T = L/T + (L/T²) * T²
L/T = L/T + L
The dimensions don't all match, hence, this equation cannot be correct.
c) ma = v2
M * L/T² = (L/T)²
ML/T² = L²/T²
The dimensions don't all match, hence, this equation cannot be correct.