Everything underwater is harder to move than it looks

Why accelerating something in water costs more than its own weight suggests.

Everything underwater is harder to move than it looks because starting, stopping, or changing the direction of an object submerged in water means dragging a surrounding shell of that water along with it, so the object behaves as though it were considerably heavier than its own mass the instant anyone tries to speed it up or slow it down. The effect is separate from drag, and physics calls it added mass.

Water that rides along with the hull

An object sitting still in air is surrounded by a fluid so much less dense than almost anything solid that the air's own inertia hardly affects how easily the object can be pushed around. Water is roughly eight hundred times denser than air, dense enough that the fluid immediately around a submerged object has real inertia of its own. Accelerating the object means accelerating that neighbouring water too, since the water against its surface has to move out of the way and close back in behind it, and the layer beyond has to shift a little to make room for the layer ahead, and so on outward until the disturbance fades some distance away. That borrowed fluid behaves as though it had been welded onto the object, adding to its effective mass for exactly as long as the object is speeding up, slowing down, or turning, and dropping out of the calculation once the object settles into a steady speed in a straight line, because water already moving steadily alongside needs no further accelerating. None of this shows up on a set of scales, since weighing the object in air captures only its own material.

A hand swept through a full sink

Sweeping a hand quickly back and forth through a full kitchen sink is added mass felt directly. The same hand waved just as quickly through open air meets almost no resistance, yet submerged and swept the same way it feels distinctly heavier to get moving, a real push against the palm that comes from the water being shoved sideways with it. Slow the hand to a gentle, steady sweep and most of that heaviness disappears, because a hand moving at constant speed is only shouldering water aside at a steady rate, which is a job for drag to describe.

A sphere carries half its own volume of water

For a sphere, the added mass works out exactly: accelerating it means accelerating an extra mass equal to half the water it displaces. A neutrally buoyant sphere, which weighs the same as the water it displaces, therefore behaves as if it were one and a half times its real mass whenever it speeds up or slows down, so a thruster sized from its weight on the scales would give it only two thirds of the acceleration expected. Other shapes vary widely with direction. A long, slender body accelerating along its own length carries very little added mass, since it barely has to push water out of its path, while the same body pushed broadside can carry far more than its own mass. That is why a torpedo-shaped hull darts forward comparatively easily yet resists being nudged sideways, and why sideways thrusters are often given more power than the geometry of the manoeuvre alone would seem to call for.

Sizing thrusters and tuning controllers

A thruster sized only against a vehicle's steady cruising speed routinely turns out to be underpowered for sharp manoeuvring, because starting from rest, stopping quickly, or changing heading means overcoming added mass on top of ordinary drag, a combined load a cruising-speed calculation never sees. Control systems have to account for the same lag. A command to change speed or direction is answered more sluggishly than the vehicle's dry weight would predict, and a controller tuned against dry-land intuition about mass tends to overshoot, correct, and overshoot again. Once the added mass is folded into its calculations the corrections settle and the vehicle stops hunting back and forth around the heading or depth it was asked to hold.

Anyone who has piloted a small underwater vehicle by joystick learns the same thing by feel: the vehicle keeps drifting for a moment after every stop command and lags for a moment after every start command. The motors and the controller are usually fine; the extra load of water accelerated with the hull on every manoeuvre appears on no spec sheet and shows up only in how the vehicle behaves once it is in the water.

Steady travel hands the problem over to drag

Added mass matters only while something is changing speed or direction, so a vehicle cruising steadily in a straight line feels none of it, and its resistance at that point comes entirely from ordinary drag. That handover, from an inertia problem during every manoeuvre to a drag problem during steady travel, is where the next article on drag in water and in air picks up. The two problems are solved by different means: added mass by sizing motors for acceleration as well as cruising, drag by shaping the hull itself.