4.3 Turning of a Bullet to Follow a Crosswind and Resulting Deflections
The point was made in Section 3.2 that a crosswind does not “blow” a bullet off course. Rather the bullet turns in the crossrange direction to follow the crosswind. This is a horizontal rotation of the bullet, and, if the bullet is to rotate horizontally, there must be a horizontal torque applied to the bullet. A horizontal torque requires a small vertical force to be applied to the bullet’s center of pressure, and this in turn requires a very small angle of attack of the bullet relative to the velocity vector. For a bullet with right-hand spin, this angle of attack must be positive for a crosswind blowing from left to right across the trajectory plane and negative for a crosswind blowing from right to left. This situation is reversed for a bullet with left-hand spin.
Figure 4.3-1 illustrates the deflection of a bullet trajectory by a crosswind. Notice that the bullet has right-hand spin, the spin angular momentum vector H is directed out the nose of the bullet, the crosswind is blowing from the left to right as the bullet flies, and the bullet trajectory curves to follow the wind. The bullet velocity vector V is exactly tangent to the trajectory, but the nose of the bullet and the spin angular momentum vector H are tilted vertically upward by a small, positive angle of attack. As explained in the preceding subsection, the principal component of the aerodynamic force (the drag force on the bullet, not shown in Figure 4.3-1) acts through both the center of pressure and center of mass, causing no torque on the bullet. However, the small angle of attack causes a small aerodynamic lift force Flift to be applied to the bullet at the center of pressure. The torque vector M, which is the vector cross product of the moment arm r and the lift force Flift, is then directed horizontally to the right of the bullet as it flies. As explained in the preceding subsection, the equations of rotational motion of the bullet cause the spin angular momentum vector H to rotate toward the torque vector M, which causes the bullet to turn to the right as it flies. This effect causes the crossrange deflection (also called crosswind drift) of the bullet, which can be large if the cross-wind is strong.
There can also be an observable vertical deflection of the bullet, in addition to the crossrange deflection. This vertical deflection is upward for the
Figure 4.3-1 Trajectory Deflection by a Crosswind (drawn for a bullet with right hand spin and a crosswind blowing from left to right
situation pictured in Figure 4.3-1. It results from the upward force Flift acting on the bullet throughout its flight. Generally, this vertical deflection is small compared to the crossrange deflection, but it can be observed, particularly in long-range target shooting.
If the crosswind blows in the opposite direction, that is, from the right to the left as the bullet flies, the bullet must turn to the left to follow the wind. This necessitates a torque vector directed horizontally and to the left as the bullet flies. Such a torque can be generated by a negative lift force (directed downward as the bullet flies), and this can happen with a small, negative angle of attack. The final result is a crossrange deflection of the bullet to the left, and a vertical deflection of the bullet downward.
If a bullet has a left-hand spin, resulting from a barrel with a left-hand twist, the spin angular momentum vector is directed out the tail of the bullet. The torque vector directions that cause the bullet to follow the crosswind then must be opposite to those for a bullet with right-hand spin. This means that the angles of attack must be opposite, with the result that the vertical deflections are also opposite in direction. These effects are summarized in the table below. The twist direction in the shooter’s gun barrel — right-hand (RH) or left-hand (LH) twist — determines whether the bullet has RH spin or LH spin. The crosswind direction is determined as the shooter looks at the target; the crosswind can blow from the shooter’s left (L) to right (R) direction, or from the shooter’s right to left direction. The crossrange deflection of the bullet will always be in the direction of the crosswind. The vertical deflection will depend on the direction of spin of the bullet.
|Barrel Twist Crosswind
|RH L to R
|RH R to L
|LH L to R
|LH R to L
Note that in this description of deflections caused by crosswinds, the effect of the yaw of repose has not been considered. This approach has been used to simplify the explanation. There always will be a yaw of repose for a bullet flying an arced trajectory. However, since all these effects are small, they can be considered as approximately additive in an algebraic sense. That is, the horizontal deflection of a bullet caused by the yaw of repose either adds to or subtracts from the crossrange deflection caused by a crosswind.