ROTATION OF THE EARTH
GENERAL
1. Although the
earth rotates at a constant rate, the correction for rotation varies with a number
of factors and, therefore, rotation is more readily considered a non-standard
condition. Factors influencing the effect of rotation of the earth on the
travel of a projectile are:
a. direction of fire,
b. A/D,
c. velocity of projectile,
d. range to target, and
e. latitude of the gun.
2. The
correction tables provide all the data needed to compensate for rotation in the
gunnery problem, however, some background theory of
rotational effects may assist in an
understanding of their application.
Rotational
Effects on Range x
3. Because of rotation of the earth, a
point on the equator has an eastward linear velocity of approximately 457 m/s.
This linear velocity decreases to zero at either pole. Consider a gun on the
equator firing due east at a target (Example 1, Figure x). During TOF of the
projectile, the gun and target will travel from G to G' and T to T',
respectively, along the circumference of the earth. The projectile, however,
travels in a vertical plane, the base of which is parallel to the original
plane of departure established at the time of firing; that is, it is pivotal to
the circumference of the earth at the gun but not at the target. At the end of
a given TOF, the projectile will be at P' when the target is at T'. Hence, the
projectile will continue along an extended trajectory and land farther east or,
in this instance, beyond its target. The normal trajectory of the projectile is
interrupted.
4. Consider the same gun firing westward
(Example 2, Figure x). Again, the projectile falls to the east of the target,
but in this instance east is short. The effect in each example is as if the QE
fired has been in error by the amount of angle "a", which is the
angle formed by the base line G' P' and a tangent to the earth at G'. When the
gun is firing eastward, angle "a" is plus (range long); when the gun
is firing westward, angle "a" is minus (range short).
5. A second effect on range is known as
projectile lag. This is best explained by use of a diagram (see Figure xx).
Assume that a projectile is fired straight up into the air, ie at an A/D of 1
600 mils. When the projectile is fired, it will have a horizontal velocity
equal to the rotational velocity of the earth. During the time that the
projectile is in flight the earth rotates moving the gun from G to G' and the
projectile moves through an arc P to P'. As this occurs in the same time, and
the horizontal velocity of the gun and projectile are the same, distance G to
G' equals distance P to P'. However, P to P' is further away from the centre of
the earth and the angle subtended is less, therefore, the round lands at
"X". Furthermore, the effect of gravity on the projectile is acting
through the centre of the earth causing the projectile to lag.
6. Tabular Firing Tables list a single
range correction for rotation of the earth that combines the rotation effect
and the lag effect. These two effects are opposing and they reach their maximum
values at different angles of departure as follows:
a. At an A/D of approximately 530 mils the rotation effect
reaches its maximum.
b. At an A/D of approximately 1 070 mils the two effects are
equal and cancel each other.
c. At an A/D of 1 600 mils, the effect of projectile lag
reaches its maximum.
Projectile
Lag
Figure
xx
7. A third consideration is the curvature
effect. Curvature effect exists because of the use of a map range for which the
surface of the earth is assumed to be flat, but the actual range is measured on
a sphere. The gun-target (GT) range is computed for a plane tangent to the
surface of the earth at the gun. When the projectile reaches this range, it is
still above the curved surface of the earth and will continue to drop,
resulting in a slightly longer true range than desired. This effect is less
than 1 meter in 1 000 meters [1] and is of little significance except at very
long ranges. It is disregarded when FTs are used, since FT ranges include
curvature effect.
ROTATION
EFFECTS ON BEARING
8. A final rotational effect is described
as the latitudinal effect. When the gun and target are at different latitudes,
the eastward rotational velocity imparted to the projectile and target is
different. For example, if the gun is nearer the equator, the projectile will
travel faster and, therefore, further to the east than the target (see Example
1, Figure xxx). The reverse is true if the target is nearer the equator.
9. When the gun and target are at the
same latitude the projectile will also be deflected away from the target. This
is because the projectile tends to travel in the plane of the great circle
containing the gun and the target at the time of firing. Because of the
rotation of the earth, this great circle plane is continually changing with
respect to its original position. As viewed from above, it would appear that
the great circle containing the gun and target is turning with respect to the
great circle followed by the projectile (see Example 2, Figure xxx). In the
northern hemisphere the latitudinal effect is to the right; in the southern
hemisphere it is to the left.
REFERENCE
[1] B-GL-306-004/FP-001, Field Artillery, Volume 6, Duties at Regimental
Headquarters and the Gun Position;
Glossary
The following symbols and abbreviations
are used in this document:
A/D - angle of departure
MV - muzzle velocity
m/s - metres per second
f - angle of elevation
g - gravitational force (9.8 m/s2)
Hv - horizontal component of velocity
Vv -
vertical component of velocity
R - range to the level point
T - time of flight to the level point
T - any given time
h - projectile height at time t.
