Which formula could be correct for the speed v of ocean waves in terms of the density ρ of seawater, the acceleration of free fall g, the depth h of the ocean and the wavelength λ?

Question 14

Which formula could be correct for the speed v of ocean waves in terms of the density ρ of seawater, the acceleration of free fall g, the depth h of the ocean and the wavelength λ?

A v = √gλ                     B v = √(g/h)                 C v = √(ρgh)                D v = √(g/ρ)

Reference: Past Exam Paper – June 2007 Paper 1 Q3

Solution:

Answer: A.

Unit of velocity: m s^-1

Consider the overall unit of the options provided.

The unit of the correct option should have the units of velocity.

Note that square root means ‘a power/index of ½ or 0.5’.

Unit of √(gλ): [ms^-2 m]^0.5 = [m²s^-2]^0.5 = m s^-1 

Unit of √(g/h): [ms^-2 m^-1]^0.5 = [s^-2]^0.5 = s^-1

Unit of √(ρgh): [kgm^-3 ms^-2 m]^0.5 = [kg m^-1 s^-2]^0.5 = kg^0.5 m^-0.5 s^-1

Unit of √(g/ρ): [ms^-2 kg^-1m³]^0.5 = [m⁴ s^-2 kg^-1]^0.5 = m² s^-1 kg^-0.5