# Acceleration

## Background

The Acceleration of objects in EVE is not based on classical physics. The 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 engine turned off you will decelerate, ultimately to a standstill. As a result, all acceleration is proportional to agility, relative to the maximum velocity of the ship, and exponential.

## How do ships in EVE accelerate and decelerate?

In terms of kinematic motion, when the ship starts to accelerate it will quickly increase its speed, but as the speed increases toward the maximum, the acceleration decreases exponentially. (In theory, the ship will never reach its top speed, however in reality, 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.
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## What decides how quickly a ship accelerates?

The idea of acceleration = force/mass can be thrown away for now, because there are only two basic attributes which determine the relative acceleration (how quickly a ship accelerates to its maximum speed): Mass and Inertial Modifier. The latter can be reduced with both skills and modules while mass can never be reduced (on the other hand, it may be increased by armor plates and active propulsion modules). The product of Mass and the Inertia Modifier 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. On the other hand, the maximum velocity of a ship does not affect relative acceleration at all. So on a nanofiber, while both the inertia and maximum speed modifier make the ship accelerate in classical terms faster, only the inertia will make its relative acceleration faster, making it align to warp faster.

To demonstrate, here's an example of this exponential acceleration:

Two ships with identical Mass and Inertial Modifier 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
Vt
Velocity after time t in m/s
Vmax
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 final term (with the exponent) gives the fraction of maximum velocity that is reached after time t. This is multiplied by the maximum velocity to find the absolute velocity at time t. Note that this only depends on time, inertia modifier and mass (e is a constant).

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 tV is the time to accelerate to velocity V in seconds. Note that at V = Vmax, 1 - V / Vmax = 0, but ln 0 is undefined, so in theory it takes infinite time to reach maximum speed (technically, the limit of tV as V approaches Vmax is positive infinity). 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.

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 Vmax 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.
```