4.4 Turning of a Bullet to Follow a Vertical Wind and Resulting Deflections
A bullet must turn upward or downward to follow a vertical wind that blows upward or downward. This is a very similar situation to a bullet turning to follow a crosswind, as described in Section 4.3, except that the direction of the wind has changed from horizontal to vertical. Figure 4.4-1 has been drawn to illustrate the conditions for a bullet flying in the presence of a vertical wind. Figure 4.4-1 is drawn for a bullet with a right-hand spin and a vertical wind directed upward. Notice that the bullet velocity vector V is exactly tangent to the trajectory. To follow the vertical wind, the bullet must rotate vertically. If the bullet is to rotate vertically, there must be a vertical torque M applied to the bullet. A vertical torque requires a small horizontal force Fvwind to be applied to the center of pressure of the bullet and to be directed to the left as shown in Figure 4.4-1. This in turn requires the spin angular momentum vector H to be rotated to the left of the velocity vector V by a very small angle. For a bullet with right-hand spin and a vertical wind directed upward, this small angle must tilt H to the left of V, so that the aerodynamic force Fvwind is directed horizontally to the left. This causes an upward directed torque vector M. As explained previously, 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 follow the vertical wind upward as it flies.
So, a vertical wind blowing upward causes an upward vertical deflection of the bullet relative to a trajectory with no wind. There also is a small horizontal deflection of the bullet, that is, a small crossrange deflection. This crossrange deflection is to the left for the situation pictured in Figure 4.4-1. It results from the horizontal force Fvwind acting on the bullet throughout its flight.
Figure 4.4-1 Trajectory Deflection by a Crosswind (drawn for a bullet
If the vertical wind is directed downward for a bullet with a right-hand spin, the bullet must turn downward to follow the wind. This necessitates a torque vector that is directed downward, which in turn requires a horizontal force directed to the right of the trajectory plane. This requires the spin angular momentum vector and the nose of the bullet to be directed to the right of the velocity vector by a small angle. The resulting vertical deflection of the bullet will be downward relative to the trajectory with no wind, and the small cross-range deflection will be to the right.
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 a vertical wind then must be opposite to those for a bullet with right-hand spin. This means that the horizontal forces must be opposite in direction, with the result that the horizontal deflections are also opposite in direction. These effects are summarized in the table below. The vertical wind direction is upward or downward, determined as the shooter looks at the target. The vertical deflection of the bullet will always be in the direction of the vertical wind. The crossrange deflection will depend on the direction of spin of the bullet as well as on the direction of the wind.
|Barrel Twist Vertical Wind||Vertical||Crossrange|
Generally, the crossrange deflection caused by a vertical wind is small compared to the vertical deflection caused by the wind, and it is seldom ever observed because vertical wind velocities tend not to be large. The vertical bullet deflections caused by vertical winds, however, are frequently seen by hunters in hilly or mountainous terrain.
In this description of deflections caused by vertical winds, the effects of the yaw of repose and crosswinds have not been considered for the purpose of simplifying the explanation. 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 a vertical wind adds to or subtracts from the crossrange deflections caused by the yaw of repose and/or a crosswind, and the vertical deflection caused by a vertical wind either adds to or subtracts from the vertical deflection caused by a crosswind.