Off-the-Break Lanes and Dead Paint

5. Off the break (OTB) lanes and "dead paint" (runner distances measured from their own start box)

This section describes off the break lanes using basic kinematics. It focuses on a single runner on a known path and a single shooter holding a fixed lane. It addresses:

When the runner reaches a given crossing point along their path. When paintballs pass through that same point. When paintballs are guaranteed not to be able to hit that runner at that point ("dead paint"). When a hit at that point is physically possible ("kill window"). How very short lane distances near the back line create a "too close" dead zone.

All distances along the run are measured from the runner’s own start box, and all ball flight distances are measured from the shooter to the same crossing point. A simple constant speed model is used throughout (constant runner speed, constant ball speed, no drag).

5.1 Two distances for one crossing point

For a single lane that intersects a single run path at one point, there are two distinct distances:

d_run: the distance along the runner’s path from the front of their start box to the lane crossing point, in feet. This is how far the runner has to travel from their start box to reach that point. d_shot: the straight line ground distance from the shooter’s barrel or bunker to the same crossing point, in feet. This is how far the ball has to travel from the shooter to reach that point.

Timing is defined so that:

Runner timing uses d_run. Ball timing uses d_shot.

For simple numerical illustrations, a symmetric one dimensional case is sometimes used where d_run = d_shot = d along a single axis. On an actual layout, d_run and d_shot can differ; the same equations apply as long as the correct measured values are used.

5.1.1 Runner distance on the layout (from the runner’s start box)

On the standard 10 × 10 ft grid:

The runner’s start box front edge is used as the origin for that runner. The runner’s typical path is a line (or curve) from this origin to their first bunker. The lane crossing point is any point where the held lane intersects that path (for example, a point along the line from the runner’s start box to a wide corner bunker).

The runner distance d_run is the path length from the start box to that crossing point. It can be approximated in several ways, for example:

Counting grid squares along the path direction (forward and left or right) from the start box to the crossing point and multiplying by 10 ft. Using a straight line approximation in the field coordinate system, such as d_run ≈ √[(Δx_run)² + (Δy_run)²], where Δx_run and Δy_run are the forward and left or right offsets of the crossing point relative to the runner’s start box.

In either case, d_run is interpreted as "X feet from the runner’s start box along their route".

5.1.2 Shooter to lane distance on the layout (ball distance)

For the same crossing point, the shooter has a separate geometry:

The shooter’s bunker (often a back center or back corner) is the shooter’s origin. The same crossing point on the runner’s path is used.

The ball distance d_shot is the straight line ground distance from the shooter to the crossing point. It can be approximated similarly:

Counting grid squares from the shooter’s bunker to the crossing point and multiplying by 10 ft. Using a straight line approximation d_shot ≈ √[(Δx_shot)² + (Δy_shot)²], where Δx_shot and Δy_shot are forward and left or right offsets relative to the shooter’s position.

This d_shot value is the distance the ball travels on the ground projection from the shooter to that lane crossing.

5.1.3 Distances on the physical field

On a real field, the same two distances can be defined and measured:

Runner distance d_run: the runner’s path from the front of their start box to the crossing point can be traced on the ground and measured with a tape or measuring wheel along that path. The measured length is d_run. Shooter distance d_shot: a straight line on the ground from the shooter’s bunker front edge to the same crossing point can be measured with a tape or wheel. The measured length is d_shot.

These two field measurements give:

d_run: how far the runner has traveled from their start box to the crossing. d_shot: how far the ball has traveled from the shooter to the same point.

5.2 Runner arrival time at the lane

Let v_player be the average runner speed along their path (for example, 15 ft/s for an average sprinter, or 20 ft/s for a fast front runner). Under a constant speed model, the time for the runner to travel from their start box to the crossing point is

t_runner,lane = d_run / v_player.

Example for a fast front runner with v_player = 20 ft/s:

Crossing at d_run = 40 ft from the runner’s start box: t_runner,lane = 40 / 20 = 2.0 s. Crossing at d_run = 60 ft from the runner’s start box: t_runner,lane = 60 / 20 = 3.0 s.

In this model, the runner cannot reach the 40 ft crossing before about 2.0 s, and cannot reach the 60 ft crossing before about 3.0 s. These times are lower bounds that follow from the assumed average speed and the measured path length from the runner’s start box.

5.3 Ball arrival times at the lane

Ball motion is modeled with:

v_ball = 300 ft/s (nominal muzzle velocity). t_shoot_start = time from the horn to the first ball leaving the barrel. This includes reaction time, raising the marker, and the first trigger pull. It is a model parameter and can be estimated or measured. Rate of fire (ROF) = 10.5 balls per second, so the spacing between shots is approximately 0.095 s.

For the first ball, the time from the horn until that ball passes the crossing point is

t_ball,1,lane = t_shoot_start + d_shot / v_ball.

For subsequent balls:

t_ball,2,lane = t_shoot_start + 0.095 + d_shot / v_ball,

t_ball,3,lane = t_shoot_start + 2 × 0.095 + d_shot / v_ball,

and so on, adding 0.095 s for each additional ball index.

Example in a symmetric one dimensional case with d_run = d_shot = 40 ft and t_shoot_start = 0.5 s:

Ball flight time to 40 ft: 40 / 300 ≈ 0.133 s. First ball time at the crossing: t_ball,1,lane ≈ 0.5 + 0.133 = 0.633 s.

For d_shot = 60 ft:

Ball flight time: 60 / 300 = 0.20 s. First ball time at the crossing: t_ball,1,lane ≈ 0.5 + 0.20 = 0.70 s.

In this example, the first ball reaches the 40 ft crossing at about 0.63 s and the 60 ft crossing at about 0.70 s. On an actual field, d_shot is taken from measurement, and the same formulas apply.

5.4 Early "dead paint" (balls that arrive too early)

At a given crossing point, a ball cannot hit this specific runner if it passes through that point before the runner reaches it. Using

Runner arrival time: t_runner,lane = d_run / v_player, First ball crossing time: t_ball,1,lane = t_shoot_start + d_shot / v_ball,

if a ball passes the crossing at time t with t < t_runner,lane, then the runner has not yet reached that location and cannot be hit there by that ball.

The interval where paint is present at the crossing but the runner has not yet arrived can be defined relative to the first ball as

t_dead,early = t_runner,lane − t_ball,1,lane.

If t_dead,early > 0, then the interval from t_ball,1,lane up to t_runner,lane is an early dead paint window for that runner at that crossing. Every ball that passes the crossing during this interval is physically too early to hit that runner at that point.

Example for a fast runner (v_player = 20 ft/s) in the symmetric one dimensional case with d_run = d_shot = 40 ft and t_shoot_start = 0.5 s:

Runner: t_runner,lane = 40 / 20 = 2.0 s. First ball: t_ball,1,lane ≈ 0.633 s. Early dead paint interval length: t_dead,early ≈ 2.0 − 0.633 ≈ 1.37 s.

With ROF = 10.5 balls per second (spacing ≈ 0.095 s):

Expected early dead paint count: N_dead,early ≈ 1.37 / 0.095 ≈ 14 balls.

In this model, approximately 14 balls can pass through the crossing before this runner reaches it, and all are dead paint with respect to that runner at that specific crossing.

5.5 Kill window (time when a hit is physically possible)

A runner’s body occupies a finite length along the path. Only while that body length overlaps the crossing region can a ball hitting that point physically contact the runner.

Let L be an effective hit zone depth along the runner’s path, in feet (for example, L ≈ 2 ft as a simple model). With constant runner speed v_player, the time needed for the runner’s body to pass entirely through the crossing region is

Δt_cross = L / v_player.

Example:

L = 2 ft, v_player = 20 ft/s gives Δt_cross = 2 / 20 = 0.10 s.

Define:

t_entry as the time when the front of the runner’s hit zone enters the crossing region. t_exit as the time when the back of the hit zone leaves the crossing region.

With a centered approximation:

t_entry ≈ (d_run − L / 2) / v_player, t_exit ≈ (d_run + L / 2) / v_player.

The kill window at that crossing is the interval [t_entry, t_exit] with duration

Δt_cross = t_exit − t_entry ≈ L / v_player.

At ROF = 10.5 balls per second (spacing ≈ 0.095 s), in a 0.10 s window the expected number of balls entering the crossing region is roughly 0.10 / 0.095 ≈ 1 (sometimes one ball, occasionally two, depending on the timing phase). In this simplified one dimensional model, only a small number of balls can physically overlap the runner’s body at that precise crossing point during the time the body is present there.

5.6 "Too close" lanes: late dead zone near the shooter

A separate dead zone arises when the lane crossing point is very close to the shooter’s position. In that case, the runner can pass that point before the first ball has time to reach it.

For a simplified one dimensional case where runner and ball share the same axis and the crossing point is at distance d from both the runner’s start box and the shooter’s position:

Runner arrival time: t_runner,lane = d / v_player. First ball time: t_ball,1,lane = t_shoot_start + d / v_ball.

If t_ball,1,lane > t_runner,lane, then the runner’s center reaches the crossing before the first ball arrives. Any paint aimed at that very close crossing is late dead paint with respect to that runner at that point, because it always arrives after the runner has passed.

The boundary distance where the runner center and the first ball reach the crossing at the same time is obtained by solving

t_shoot_start + d / v_ball = d / v_player

for d. Rearranging,

t_shoot_start = d × (1 / v_player − 1 / v_ball),

so the minimum effective lane distance is

d_min = t_shoot_start / (1 / v_player − 1 / v_ball),

or equivalently,

d_min = t_shoot_start × (v_ball × v_player) / (v_ball − v_player).

For distances d < d_min in this model, the runner’s center always passes the crossing before the first ball can physically arrive there.

Example A (fast runner, v_player = 20 ft/s; t_shoot_start = 0.5 s; v_ball = 300 ft/s):

d_min = 0.5 × (300 × 20) / (300 − 20). d_min = 0.5 × 6000 / 280 = 3000 / 280 ≈ 10.7 ft.

In this simplified picture, a lane whose center is closer than about 10.7 ft along that axis is too close. The runner’s center reaches it before the first ball can arrive.

Example B (average runner, v_player = 15 ft/s; same t_shoot_start and v_ball):

d_min = 0.5 × (300 × 15) / (300 − 15). d_min = 0.5 × 4500 / 285 = 2250 / 285 ≈ 7.9 ft.

For the slower runner, the minimum effective lane distance is slightly shorter, about 8 ft.

On an actual field, each crossing point has a specific d_run from the runner’s start box and d_shot from the shooter’s bunker. Comparing the actual t_runner,lane and t_ball,1,lane for those measured distances indicates whether the crossing behaves like a too close late dead zone for that runner at that point, a region with an early dead paint window, a short kill window, and a late dead window, or a distant crossing whose practical effect is limited by other factors such as gravity, drag, and intervening bunkers.

5.7 Early and late dead zones at a fixed crossing

For a given crossing with fixed d_run and d_shot, relative to one runner and one shooter:

Early dead zone: Consider the time t_ball,1,lane when the first ball reaches the crossing and t_entry when the runner’s body begins to overlap the crossing. If t_entry > t_ball,1,lane, then the interval [t_ball,1,lane, t_entry) is an early dead paint window. Any paint that passes the crossing during this interval is early relative to the runner at that location.

Kill window: The interval [t_entry, t_exit] is the kill window. During this period, the runner’s body overlaps the crossing region and hits are physically possible.

Late dead zone: After the runner’s body has cleared the crossing (t > t_exit), any paint that passes through the crossing is late relative to that runner at that point and is again dead paint in this one dimensional model.

Adjusting the crossing distance relative to the shooter and runner changes these regions:

Very close crossings can behave as late dead zones if t_ball,1,lane is always later than t_exit. Crossings farther downfield can still provide a kill window, as long as t_ball,1,lane is not excessively delayed and line of sight is maintained.

5.8 Simplified descriptive summary

In summary, the runner’s motion is described from the front of their own start box to a specific point on their path. That distance is d_run and gives t_runner,lane = d_run / v_player. The ball’s motion is described from the shooter’s position to the same point. That distance is d_shot and gives t_ball,1,lane = t_shoot_start + d_shot / v_ball.

Comparing these times reveals:

Intervals before the runner arrives where paint at that crossing is early and therefore dead for that runner. A short interval while the runner’s body overlaps the crossing region, during which hits are physically possible. Intervals after the runner has passed where paint at that crossing is late and again dead for that runner.

In very short lane distances near the back line, the runner can pass the crossing before the first ball can arrive, creating a too close dead zone. At moderate distances, early dead paint, a brief kill window, and late dead paint all appear. At larger distances, gravity, air resistance, field obstacles, and loss of line of sight become the dominant limiting factors.

This framework describes off the break lanes purely in terms of distances, speeds, and timing, with all runner motion measured from the other player’s starting position and all ball motion measured from the shooter to the same physical point on the field.