Check dimensions of the formula and verify that the formula

Check dimensions of the formula and verify that the formula is dimensionaly correct :

v = sqrt(g + (p / P))

v - velocity, g - gravity acceleration, h - size, p - pressure, P - density

Solution

v = sqrt(g + (p / P))

v - velocity, diamension of Velocity   = length travelled per unit time=[L]/ [t]

g - gravity acceleration, (gravitational Acceleration) (g) has a dimension of LT^-2 where L is length, and T is time.

h - size, has a dimension of L

p - pressure, Dimensional Formula of Pressure = M¹L^-1T^-2

P - density, Dimensional Formula of Density =M¹L^-3T⁰

according to the given formula, v = sqrt(g + (p / P))

[L]/ [T] = sqrt( LT^-2 + (M¹L^-1T^-2 / M¹L^-3T⁰)) = sqrt( L/T² + (L² / T²)) = sqrt( L)/sqrt( T) = [L]/ [T]

becouse on adding two length we get a length and sqrt of any length is also a length this same for time .

hence the given formula is dimensionaly correct