5.3.2
Crosswinds
When a bullet is fired in a crosswind, its velocity relative to the moving air mass has a small component in the crossrange direction, as well as a main component in the downrange direction toward the target. Because of this small crossrange component of relative velocity, there is a small crossrange component of air drag force which causes the bullet to deflect in the direction of the wind. It is convenient to think of the crosswind tending to "drag" the bullet along with it. However, because of its inertia, the bullet does not follow the crosswind precisely. The crossrange bullet motion is accelerated relatively slowly, and in fact the crossrange component of the bullet's velocity never does grow to equal the crosswind velocity.
Figure 5.32
is an example of these effects. The 200 grain .308 HPBT MatchKing bullet is fired straight downrange with a 2700 fps muzzle velocity and a 5 mph crosswind. If the bullet precisely followed the wind, it would have the path represented by the dashed line in the figure, and after traveling 1000 yards it would have a crossrange deflection of about 137 inches. But in reality, the bullet follows the true path shown in the figure, and at 1000 yards it is actually deflected about 39 inches. While the 39 inch deflection certainly requires a windage correction, it still is a lot less than the 137 inches.
A mathematical equation has been derived for the deflection of a bullet by a crosswind. In the case of level fire (rifle barrel level) and without a headwind, this equation is
where
Z
is the crossrange deflection (in inches)
V
ew
is the crosswind velocity (in inches/second)
(1 mph = 17.60 inches/second)
t
is the bullet true time of flight (in seconds)
X
is the range from shooter to target (in feet)
V
o
is the muzzle velocity (in feet/second)
The quantity inside the parentheses in this equation has been called the lag time. This name came about in the following way. The quantity
t
is the total time of flight of the bullet. The quantity
X/V
o
can be recognized as the time of flight the bullet would have if it were fired in a vacuum. In a vacuum there would be no air drag to slow the bullet down, so the time of flight would be the level distance
X
to the target divided by the level component of velocity, which would be the muzzle velocity
V
o
. Then the difference between the true time of flight and the vacuum time of flight is the delay, or lag, caused by the air drag slowing the bullet down.
It is also interesting to examine the equation for the crossrange component of the bullet velocity. This equation is
where
V
z
is the velocity of the bullet in the crossrange direction
V
x
is the velocity of the bullet in the downrange direction
V
ew
is the crosswind velocity
V
o
is the muzzle velocity
The downrange velocity component
V
x
is equal to the muzzle velocity when the bullet leaves the muzzle, and then it continually decreases as the bullet flies down the range and air drag slows it down. The quantity in the parentheses in this equation can be regarded as a "velocity lag factor." This quantity has a value between zero (at the muzzle where
V
x
= V
o
) and one (when
V
x
drops to zero). In practical shooting situations
V
x
never approaches zero, and so
V
z
never grows to be as large as the crosswind velocity
V
ew
.
We can apply these formulas to the example in
Figure 5.32
to see how they work. The crosswind velocity
V
ew
is 88.0 inches/second (5 mph). For a range of 1000 yards, our computer program calculates the true time of flight to be 1.552759 seconds and the remaining downrange velocity
V
x
to be 1396.0 fps. If the bullet exactly followed the wind, in other words if it had the full crosswind velocity, then the deflection at 1000 yards would be
Z = V
ew
t + 88.0 x 1.552759 = 136.64 inches
This is the coordinate of the upper dashed line in
Figure 5.32
. The range
X
is 3000 feet and the muzzle velocity
V
o
is 2700 feet per second. The vacuum time of flight would be
Then, the lag time caused by the drag is
The true deflection of the bullet at 1000 yards is
The velocity lag factor is
And the bullet's crossrange velocity
So,
V
z
is less than half the crosswind velocity even after the bullet has traveled 1000 yards.
These equations are important for understanding crosswind effects, but they are not particularly useful to the shooter because it has not been a practice historically to compute and list bullet time of flight in ballistics tables. Even if time of flight were listed, the shooter would still have the inconvenience of calculating the crosswind deflections he needed from the equation. For this reason, we decided to calculate and publish the crosswind deflections for all Sierra bullets, rather than the times of flight. These deflections are listed in the Ballistics Tables of the Manual.
