1. Pushing a ship moves the water too
If you move your hand through water in a bath, it feels heavier than moving it through air. This is because your hand pushes water aside and moves some of it along [4]. The same thing happens with ships. When a ship accelerates or slows down, it must move some of the surrounding water as well. This makes the ship's resistance to motion (inertia) appear to increase [4].
Scientists call this extra resistance "added mass" or "virtual mass." When calculating how a ship moves, engineers add the added mass to the ship's actual mass to find the total mass used in the equations of motion [2].
2. Why water matters more than air
Even in the air, moving an object pushes the air around it. However, added mass is rarely a big topic for everyday objects in air. A summary of a thesis from Hokkaido University states that in dense water, the added mass appears about 800 times larger than it does in air [1]. Because of this, it must be carefully considered when designing ships or studying their movement [1].
This "800 times" figure compares how added mass shows up in water versus air. It does not mean the ship itself becomes 800 times heavier. It means the added mass in water is about 800 times that in air.
3. The discovery came from a pendulum, not a ship
Although we talk about ships now, the idea started with pendulums. When a ball on a pendulum swings through air or water, just looking at the ball's weight and buoyancy is not enough to explain its motion. In 1828, Bessel, a German scientist, studied pendulum motion. He pointed out that you must consider not just the fluid's buoyancy, but also the fluid's inertia [3]. He described this increase in inertia as adding a mass equal to k times the mass of the fluid that was pushed aside [3].
Some papers mention that Du Buat found as early as 1786 that the motion of a ball vibrating in water could not be explained without including added mass [2]. However, this date and name come from secondary descriptions, so they are treated with caution.
4. How the numbers changed for a ball in air and water
Bessel swung a ball slightly more than 2 inches (about 5 cm) in diameter through air and water to measure the coefficient k. According to Stokes' description, the value in air was 0.9459 (later revised to 0.956). In water, the value was about two-thirds of the value in air [3].
These numbers are multipliers relative to the mass of the fluid pushed aside. The number is smaller in water than in air, but because water is much heavier than air, the actual increase in inertia is larger in water. In 1851, Stokes calculated the added mass of a sphere theoretically. He showed that for a sphere, the added mass is about half the mass of the fluid it displaces [4].
5. How much heavier does a real ship feel?
A paper on towing experiments during the acceleration stage of ships states that a ship's added mass can reach up to 30% of the ship's own mass [2]. For example, if a ship has a mass of 10,000 tons, 30% would be 3,000 tons. This is an upper-limit estimate; it is not always this large.
The value of added mass changes depending on the direction of movement: forward, sideways, or rotating [4]. The 30% figure also depends on which direction is being considered.
6. Using added mass to calculate docking
When a ship approaches a quay, the sum of the ship's mass and its added mass is called "virtual mass." This is used to calculate the energy of the ship hitting the dock. A 1987 paper by Ueda defines this virtual mass as the sum of the berthing ship's mass and its added mass [5].
In a 1977 experimental report by the Port and Airport Research Institute (then the Port Research Institute of the Ministry of Transport), virtual mass was expressed as a linear function of the ship's displacement. When the ratio of water depth to draft was 1.10, the virtual mass coefficient was about 2.3. This was larger than the design standard of 1.5 to 1.6 at the time, showing that the ratio of water depth to draft matters [6]. This data is from 1977 and is not the current standard.
7. Try comparing water resistance at home
In the bath, try moving a ladle or a thin board through the water. How does the feeling change when you move it slowly versus quickly? Try changing the angle of the ladle or board to compare the direction where water escapes easily with the direction where it escapes less easily.
Because wet floors are slippery, sit on the edge of the bathtub to do this. Please do this activity with a parent or guardian to ensure safety [1].
If you want to read more, look at the thesis summary from Hokkaido University [1] or the 1977 report from the Port and Airport Research Institute [6].