Acceleration

Background

EVE's physics engine is based on a 'fluid dynamics model' which assumes that 'space' has some substance to it and thus some friction, this means that with the ship engine turned off you will decelerate, ultimately to a standstill.

How do ships in EVE accelerate and decelerate?

When a ship starts to accelerate it will quickly increase its speed but as the speed increases the acceleration decreases exponentially which means that, in theory, the ship will never reach its top speed. In reality however, it won't take long to get so close to top speed that the difference is negligible and EVE rounds up the figure on your display.

Deceleration is simply acceleration in a direction opposed to the one you are travelling in, I.E. 'braking'. The closer to your top speed you are the faster you will decelerate, again, in theory you can never completely stop the ship.

Please note: Although we talk here about ships the mechanics apply to any object moving under power in the EVE universe, for example missiles and drones.

What decides how quickly a ship accelerates?

There are two attributes which determine how quickly a ship accelerates; mass and Inertial Modifier. The product of these gives the ship's agility which determines how quickly the ship accelerates (and thus how quickly it turns); lower values imply better acceleration and turning speed.

Two ships with identical masses and Inertial Modifiers but different top speeds will reach their respective top speeds in the same period. This means a faster (greater top speed) ship will have a higher acceleration in ms^-2 but will take the same time to reach, for example, the speed required to use warp engines.

The Mathematics

The following formula, assuming the ship is starting from standstill, decribes how acceleration works in EVE:
Vt = Vmax * (1 - e^(-t * 10^6 / (I*M) )

Where:

Vt = Velocity after time "t", metres per second.
Vmax = Maximum velocity (top speed), metres per second.
e = Mathematical constant (base of the 'Natural logarithm' - ln).
t = Time, seconds.
I = Inertia Modifier, dimensionless.
M = Mass, kilogrammes.

It can also be written as:
T = I*M * 10^-6 * -ln (1 - Vt / Vmax)

Where:

T = Period to reach Vt, seconds.
I = Inertial Modifier, dimensionless.
ln = Standard mathematical function, Natural logarithm.
Vt = Target velocity, metres per second.
Vmax = Maximum velocity, metres per second.

Please note: Strictly speaking, velocity is a vector, so it has both direction and magnitude but we're really only interested in its absolute value (I.E. the magnitude part) more commonly called 'speed'.

Example: Pete has just got himself a new freighter, a Charon.
The Charon has a Mass of 1,200,000,000 kg and an Inertia Modifier of 0.02176875 (after adjustment for skills), he wants to know how long it takes for his ship to reach the speed needed to enter warp (75% of the ship's top speed). Pete puts the values into the formula above:

Time to Warp = 0.02176875 * 1.2 * 10^9 * 10^-6 * -ln (1 - 0.75 / 1)

                   = 0.02176875 * 1.2 * 10^3 * -ln (1 - 0.75)
                   = 0.02176875 * 1200 * -ln 0.25
                   = 26.1225 * 1.38629436
                   = 36.2134744 seconds.

In case you are wondering, the "0.75 / 1" for "Vt / Vmax" is just a shortcut because we know that the ratio of the two is 0.75, since we want 75% of whatever value the maximum is.