6.6
**
Other Useful Equations
**

As a final note to this section, we will provide some other equations which are very useful for trajectory calculations.

*
Velocity Required to Reach a Specific Range
*

When trajectories are calculated, it is customary for the trajectory parameters (drop, time of flight, remaining velocity, etc.) to be provided at specific ranges from the muzzle (100 yds, 200 yds, etc.). Consequently, we need an equation which gives the pseudovelocity required to reach a specific range point. This equation is:

**
(6.6-1)
**

where
**
x
**
**
2
**
is the required range to be reached and
**
u
**
**
2
**
is the pseudovelocity at that range point. Equation (6.6-1) applies in all pseudovelocity subranges except the lowest velocity range in which k = 9. In that subrange the correct equation is:

**
u
**
**
2
**
**
= u
**
**
1
**
**
exp(-
**
**
f
**
**
) (6.6-2)
**

where

**
(6.6-3)
**

*
Drop
*

Drop is defined to be the vertical difference between the trajectory vertical position
**
y
**
and the extended bore centerline:

**
d = y - x tan
**
**
q
**
**
o
**

This definition applies at every range point
**
x
**
and for any firing elevation angle X. According to this definition drop is always a negative number, signifying that the trajectory curves downward from the extended bore line.

*
Determination of True Muzzle Velocity
*

The velocity of a bullet always is measured a few yards downrange from the true muzzle position. If the chronograph is too close to the muzzle, then erroneous velocity values are caused by effects such as muzzle flash or powder gases. Because the chronograph is downrange from the muzzle, we need an equation which gives the true muzzle velocity based on the velocity measured by the chronograph. This equation is:

where
**
v
**
**
m
**
is the true muzzle velocity,
**
v
**
**
c
**
is the velocity measured by the chronograph, and
**
D
**
**
mc
**
is the distance between the gun muzzle and the
*
center
*
of the two screens of the chronograph.

For the special case in which
**
k
**
= 9, equation (6.6-5) reduces to: