Question

Salt water is denser than fresh water. a ship floats in both fresh water and salt water. compared to the fresh water, the volume of water displaced in the salt water is

Answer 1

The ship floats in water due to the buoyancy Fb that is given by the equation:

Fb=ρgV, where ρ is the density of the liquid, g=9.81 m/s² is the acceleration of the force of gravity and V is volume of the displaced liquid.

The density of fresh water is ρ₁=1000 kg/m³.

The density of salt water is in average ρ₂=1025 kg/m³.

To compare the volumes of liquids that are displaced by the ship we can take the ratio of buoyancy of salt water Fb₂ and the buoyancy of fresh water Fb₁.

The gravity force of the ship Fg=mg, where m is the mass of the ship and g=9.81  m/s², is equal to the force of buoyancy Fb₁ and Fb₂ because the mass of the ship doesn't change:
 
Fg=Fb₁ and Fg=Fb₂. This means Fb₁=Fb₂.

Now we can write:

Fb₂/Fb₁=(ρ₂gV₂)/(ρ₁gV₁), since Fb₁=Fb₂, they cancel out:

1/1=1=(ρ₂gV₂)/(ρ₁gV₁), g also cancels out:

(ρ₂V₂)/(ρ₁V₁)=1, now we can input ρ₁=1000 kg/m³ and ρ₂=1025 kg/m³

(1025V₂)/(1000V₁)=1

1.025(V₂/V₁)=1

V₂/V₁=1/1.025=0.9756, we multiply by V₁

V₂=0.9756V₁

Volume of salt water V₂ displaced by the ship is smaller than the volume of sweet water V₁ because the force of buoyancy of salt water is greater than the force of fresh water because salt water is more dense than fresh water.