## Friday, November 20, 2015

### How You Can Improve Bumping by Studying a Desk Gadget: Part I - The Theory

Bumping ships in EVE is fundamentally about exchange of energy between objects.  Mechanical systems store energy kinetically, or in some potential, such as gravity or a spring.  Now, in ship bumping, there are no explicit springs because bump events are practically instantaneous in EVE.  So the difference we have to focus on is the mass of the ships.

Energy flows most efficiently from one object to another when the masses are the same.  That is, matching the form of a source of energy to its destination, improves the efficiency of the conversion.  Unlike most computer games, the world of EVE is based on a physical model that enforces these laws, so understanding them deeply is the source to doing things others can not.

In this post I describe a bumping technique that has been used in EVE primarily for bumping capital ships, but the principle appears to apply to all ships.  Black Legion made use of this technique when bumping capitals out of station docking radius No doubt, other groups use this approach as well, but I have yet to see it described publicly or with insight.  I hope you enjoy these notes.

### In a nutshell...

The case of one ship bumping another is easy enough to understand.  Both energy and momentum are conserved for object collisions in EVE.  While it is reassuring to know that all the energy you put into the collision comes out as ship motion, the problem is that much of that energy is reflected in the bumping ship as it bounces off the target ship.  That is, the energy transfer between the objects was limited by the fact that the incident bumping ship bounced backwards.  I described the physics of this system in Part III of my original notes if you are interested in data and analysis supporting this claimA good system to help build intuition for what is happening in an elastic collision model is a common desk gadget, the Newton's Cradle.

 Figure 11-1: Animated Newton's Cradle.  Used under Creative Commons from Lurlock:Wikipedia.
If you've ever played with one of these gizmos, you'll know that energy is mostly conserved when you raise one of the spheres and start the pendulum collisions.  Bumping between two ships in EVE is, in most respects, identical to the effect of a Newton's Cradle with two spheres that that are proportioned to the ship masses.*  All the energy that you put into the collision comes out as either motion in the target ship or reflected motion in the bumping ship.

The fundamental insight of this post is that because energy exchange between objects in EVE follows a fundamental physical law, the real-world technique of impedance matching applies to in-game bumping interactions.

The way in which EVE motion is different, of course, is that all ships are experiencing drag proportional to their velocity so after being bumped, the target ship will slow down, provided it was stopped to begin with.  If you want more detail on how to calculate ship bumping effects, you can review Part III of my notes in my September 15th blog post.** On with an example...

### Two Ships, One Bump

Lets consider bumping a target ship, that has mass m1 and inertia, I, with a bumping ship that is going velocity v1, and also has mass, m1A quick thought experiment based on the Newton's Cradle tells us exactly what will happen here -- the bumping ship will stop in its tracks, transferring 100% of its energy to the bumped ship!  In the figures below, I colored the bumping ship in a rust color for obvious Minmatar reasons.

 Figure 11-2: Bumping of ships of identical mass.  All of the energy is transferred to the target ship.

Based on the elastic collisions, and for m2 = m1,
$\large v_2(t = 0^+) = \frac{2v_1(t=0^-) m_1}{m_1 + m_2} = v_1(t=0^-)$

If you try this in game using the 'Approach' command you will notice that your bumping ship continues to move forward, but this is because your motion command is still activeIf the motion command were removed at the moment of collision your ship would stop dead in place and the target ship would take your velocity and drift a distance exactly $x=v_1m_2I_2 = v_1\tau_2$Energy is obviously conserved in this instance because the energy of the target ship is equal to the energy that the bumping ship had at the start of the collision event.  This is identical to the Newton's Cradle case.

Now lets consider another common case, where the bumping ship is less massive than the target ship, or m1 < m2. In this general case, the velocities after bumping are,
$\large v_1(t = 0^+) = v_1\frac{m_1-m_2}{m_1 + m_2}$
$\large v_2(t=0^+)=v_1\frac{2m_1}{m_1 + m_2}$

As you can see from the equations above, when the mass of the target is greater than the bumping ship, the bumper will be reflected back with some velocity.  You can try this at home, either with your metal balls, or with ships in EVE.
 Figure 11-3: When bumping a larger ship, a lot of the bumping energy is reflected.

Looking at the velocity of the target ship, you can see that more of your velocity is transferred to the target when the masses are close togetherIs there a way to exchange energy between more similar masses making the energy exchange ratio between the ships more complete

### Let's try an Experiment...

What if we introduce an intermediate ship that is in-line with the bumping ship, placed very close to the target ship and is stationary.  Instead of bumping the target, instead I aim to bump the intermediate ship, having some mass m2which will then strike the target.

Crazy talk!  Or is it...?

I've draw this case out below, and you can imagine from the intuition we developed above that the bumping ship, m1, is reflected with less velocity if strikes m2, which has less mass, than if it had bumped into m3, which has greater mass.

 Figure 11-4: An intermediate ship is included to improve energy transfer from the bumping ship to the more massive target ship.

If I analyze this by writing the m3 target ship's velocity after both bump events are completed,
$\large v_3(t=0^+) = v_1\frac{4m_1m_2}{(m_1 + m_2)(m_2+m_3)}$

What mass should we choose for m2?  Can this approach really be better than bumping m3 with m1? Before I write the ratio of final velocities, lets write these mass arrangements as ratios so we can get it down to one number.

Define the ratio of the target ship, m3, to the bumping ship, m1, as K = m3/m1, and let's choose the intermediate ship mass, m2, to be a ratio Î± of the bumping ship, m2 = Î± m1Writing the ratio of the target bump velocities for these cases is,

\large \begin{align*} \frac{v_{3,3ships}(t=0^+)}{v_{2,2ships}(t=0^+)} & = \frac{2m_2(m_1 + m_3)}{(m_1+m_2)(m_2+m_3)} \\ & = \frac{2 \alpha (1 + K)}{(1+\alpha)(\alpha + K)} \end{align*}             (11-2.x)

We can find the optimal value of the intermediate ship, m2 = Î± m1, by looking for where equation (11-2.x) has a minimum point and then solving for Î±It turns out that the optimal mass for the intermediate ship is simply the square root of the ratio between the bumping ship and the target ship, or,
$\large \alpha_{optimal} = \sqrt{K}$                                                  (11-2.y)

Therefore, the optimal mass of the intermediate ship, m2

\large \begin{align*} m_{2,optimal} & = \alpha_{optimal} m_1 \\ &= \sqrt{K} m_1\\ &= \sqrt{m_1 m_3} \end{align*}

This quantity is the geometric mean of the bumping and target ships.  It is certainly nice when I get a simple formula that I can remember.  Let's see how this performs in a simple exampleIf I'm bumping a BS (100Mkg) with an MWD cruiser (15Mkg), then K = 100/15 = 6.6, which means that m2 = Î±opt m1 = 38Mkg.  Even if I could find or fit a ship with a mass near 38Mkg, would it really help?  Solving equation (11-2.x) I find that using the optimal intermediate ship will increase the bump distance of this BS target by about 20%.  This is certainly helpful but was all this work really worth an extra 20%?

If I plot equation 11-2.x for a wide range of mass ratios, K, I find the benefit in bump velocity (and corresponding distance) below in Figure 11-5.  As you may expect, this helps us more if the ship masses are very different, i.e. bumping something very massive with a much smaller ship.  Considering ships in EVE span a mass range of over 1000, I can see how this would be extremely helpful when bumping capital ships.

 Figure 11-5: Benefit of using one intermediate ship with optimal mass to bump a ship that is K times more massive.  If you are in a ship with the same mass as your target there is no benefit, as with the equal-mass Newton's Cradle.

The other question pilots should ask is, "What masses of intermediate ships are needed for various bumping conditions?  Are ships with these masses available?"  Indeed, the optimal masses are frequently difficult to synthesize in the game.  There are, however, some important bumping cases where CCP has given us just the right tools for the job.

### The Classic Match-up

Lets consider capital bumping conditions -- a 500MN Vagabond (61Mkg) trying to bump a carrier (1163Mkg). Intuitively, the bump cruiser is going to bounce off something weighing almost twenty times as much unless we give it some help.  Figure 11-6 below gives a sense of the mass difference if these were solid spherical elements in a Newton's Cradle.

 Figure 11-6: As an aid to intuition, I have included a Newton's cradle with a carrier and a bump-Vagabond sized for spheres with constant density. When the target ship and bump ship are very different masses, the bump ship will bounce due to reflected energy.

Based on the work I showed above, an intermediate ship should get us 40% greater bump distance for the same bumper velocity.  The desired intermediate mass is 266Mkg, which is an unusual ship mass.  I wouldn't deign to mine in EVE, but those of you who do will immediately recognize this as being very close to the mass of the Orca, the ORE industrial capital ship.  Who knew that these ships had a purpose other than carebear activities!

 Figure 11-7: Here is a Newton's Cradle that you might find on a station undock in low-sec. By choosing an intermediate that has both optimal mass and a carebear demeanor, you can vastly improve your bump results.

Even more beneficial, consider the case where you need to bump this same capital ship with a standard MWD cruiser, i.e. m1 =  15Mkg.  In this case, the optimal intermediate mass is close to m2 = 130Mkg, which can be closely synthesized with a BS hull but anything in the 100Mkg range will do a nice job of increasing the bump distance over 60%!

Does it make you wonder whether early CCP engineers chose mass ratios in the game to achieve such benefits for clever players?

I have taken data on these effects and I look forward to sharing it with the community.  I will wait to do so until December so that I can prepare all of these materials in one form.  Stay tuned -- there will be a couple important surprises.

### How does this work, you ask?

Consider for a moment energy exchange in the real world.  When an object with mass is moving, it has stored kinetic energy A very large mass doesn't need to move as fast to have the same energy as something that is small.  When we transfer mechanical motion to another object, it is temporarily stored in another form.  We can think of this intermediate state as the potential energy of a lossless spring

A thought experiment to aid in this: Imagine the moment when the bumping ship strikes the spring, storing all of it's energy, and coming to a stop.  The force applied by the spring on the bumper is the same as on the target ship.  The two ships, however, accelerate differently away from the event, according to Newton's law,

$\large a_i = \frac{F_{spring}}{m_i}$

When we add an intermediate ship whose mass is in between the bumping ship's mass and the target's mass, we improve the ratio of energy that is transmitted forward to energy reflected.  Each intermediate mass of this type helps to match the mass of the energy source to the target.

This principle is the same for many systems that transfer energy from one form to another, or transmit energy over a distance.  For example, an efficient loud speaker interfaces between the motion of a linear magnetic driver and acoustic waves in a large space with a tapered horn.  The horn of a loudspeaker accomplishes this by transitioning from the linear motion of the driver which moves a small amount of air a long distance, to a large amount of air a short distance.  This is the idea of impedance matching, and bumping in EVE also exploits this mechanism.

Efficient electrical systems also need to match between the ratio of electric and magnetic fields to transfer energy from one place to another.  This is the principle behind impedance matching for antennas or cables.  From this perspective, you can imagine that having a continuum of matching states between the objects should enable you to transfer all of the bump energy to the target!

 Figure 11-8: Hypothetically, a large number of ships can be used in a taper configuration to transfer 100% of the bumper energy to the target.  I think GoonSwarm could try something like this.

The theory for a large number of ships (or a continuum) can be generalized from what I have shared in this post.  I already tweeted a hint to the solution!  I will try to add this analysis to the data notes as an extension for enthusiastic readers.

I hope that your bumping experiences are greatly improved by considering energy exchange in mechanical systems.

### Two final notes

Don't hesitate to use knowledge of game physics as part of your pilot admissions criteria for corporations and alliances.  Be sure to make applicants answer at least one trick questions about mechanics before admitting them.  This may not help you weed out spies but it certainly is fun.

Finally, getting good bump data in game is very tedious.  It takes a carefully configured setup to make reproducible results for ship bumping.  This is because any misalignment will result in a bump trajectory that requires angle measurement to correlate with theoretical 1-D predictions.  Furthermore, the large time-constants of the ships we are discussing in this post means accurate alignment can take a lot of time. Real combat situations on Tranquility are likely setup rather hastily, but I hope this post has shown you that you can always benefit from studying physics to improve your bumping performance.

#### Interlude

This post is brought to you by Physics and liquid DnB.  Enjoy responsibly.

"If what you have done yesterday still looks big to you, you haven't done much today."
- Mikhail Gorbachev

* - There are details in the analysis of real Newton's Cradles that I am ignoring here, including compressibility of the steel, proximity of the balls immediately prior to contact, and change in the contact area during the collision.  I am confident that these can be ignored without detracting from the intuition that studying this system can provide.  I could alternatively chosen to study billiard ball collisions for this model but this also has higher-order considerations like backspin which would also require this footnote.

** - Intuitively, you could imagine increasing the drag on the objects by submerging your Newton's Cradle in a tub of a viscous fluid but that could get messy... and how would you transfer energy to the first collision without drag?  So, it could get tricky but you get the idea.

Disclaimer
I want to reiterate that my work on energy and momentum conservation in EVE ship interactions fits the data that I have, however, I have not actually inspected any of the game code.  Former CCP developers have remarked on the accuracy of my work, but like everything else, readers should seek to develop their own understanding based on as much information as they can gather. Your mileage may vary, but it shouldn't.