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.

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 an Inertial Modifiers, both of which can be helped using an Inertia Stabilizer on your ship. The product of mass and the inertia stabilizer values 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. Thus, a ship with a higher top speed will have a higher acceleration in ms^-2 but will take the same time to reach the speed required to use warp engines.

The mathematics

The following formula describes the velocity of a ship accelerating from a standstill after a particular time:

File:Accel-formula-Vt.png

where:

t 

Time in seconds

V_t 

Velocity after time t in m/s

V_max 

Ship's maximum velocity in m/s

I 

Ship's inertia modifier, in s/kg

M 

Ship's mass in kg

e 

Base of natural logarithms

Explanation: The term after the multiplication sign is the fraction of maximum velocity which is reached in time t. Note that this depends only on time, inertia modifier and mass (e is a constant). This is multiplied by the maximum velocity to find the actual velocity at time t.

The 10⁶ term cancels out a factor of one million in the mass term. So to simplify you can ignore the 10⁶ and use the mass of the ship in millions of kg instead of kg.

As for acceleration itself, this is just the first derivative of velocity with respect to time.

Rearranging the formula for t we arrive at the formula for time taken to accelerate from zero to V:

File:Accel-formula-tV.png

where t_V is the time to accelerate to velocity V in seconds. Note that at V = V_max, 1 - V / V_max = 0, but ln 0 is undefined, so in theory you can never reach maximum speed. In practice the game simulation is not perfectly accurate and it actually takes finite time to reach maximum speed to within whatever precision the simulation uses.

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. Since this is 75% of the ship's top speed regardless of what that top speed actually is, he doesn't bother calculating it, but instead simplifies by substituting 0.75 and 1 for V and V_max respectively.

Time to Warp = 0.02176875 × 1.2 × 109 × 10-6 × -ln (1 - 0.75 / 1)
                   = 0.02176875 × 1.2 × 103 × -ln (1 - 0.75)
                   = 0.02176875 × 1200 × -ln 0.25
                   = 26.1225 × 1.38629436
                   = 36.2134744 seconds.