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film and record their coordinates. If both the reference line and the line on the missile are the finest lines resolvable, the standard error in the distance from the reference to the missile will be equal to A/2/20 the width of each line (this allows for errors on both measurements). Expressed mathematically
Af = V2Z0/20 (2)
where Af is the standard reading error in millimeters on the film and w again is the width of the finest resolvable line in millimeters.
By substitution of Eq. 1 in Eq. 2
Af = V2/40# (3)
If the camera is to be used as a tracking camera, Eq. 3 is sufficient because if the resolving power is too low to give sufficient reading accuracy, a longer lens can be used so that reading errors result in correspondingly less error in missile position determination.
If the camera is not used to track the missile but has a fixed coverage, then reading accuracy must be considered in terms of the coverage desired with this camera. Essentially, the data desired are acceleration values, but since these are obtained by differencing two velocity values which in turn are obtained by differencing two position values, it is always necessary to obtain two more frames including the missile than the number of acceleration values desired from this one camera. Also the relation between reading accuracy on the film and acceleration accuracy must be defined.
If only position errors are considered it can be shown that
where As is the standard position error
of the missile in feet, A a is the standard acceleration
error in feet per second
squared, As is the standard position error
of the missile in feet, and
T is the time between successive
frames in seconds.
But if the width of the frame in the direction of rocket travel is Wf in millimeters and the distance in feet in space that this film covers is Ws, projecting film readings into space:
Wf/W. = Af/A.
(5)
Substituting Eq. 5 and Eq. 3 in Eq. 4 : Aa = V3 W./2QWfRT* (6)
Ws, the distance covered along the trajectory, is a function of the focal length of the camera and the distance from the camera to the missile, but since, n general, the focal length and camera distance can be varied, it is more useful to consider the distance in terms of the coverage desired. If the average velocity of the missile as it passes the camera is F and S is one less than the number of frames in which the missile is exposed, then:
W. = VST
(7)
where T again is the time between successive frames.
If Eq. 7 is substituted in Eq. 6
A. = V3~ VS/20 WfRT (8)
Since there must be two more frames than the number of acceleration values and S is always one less than the number of frames in which the missile is exposed, S is always one greater than the number of acceleration values. If Eq. 8 is written in terms of both the acceleration error and the number of acceleration values desired and converted to G's,
Aa/S = V3 F/640 WfRT (9)
Since Frame Speed (F.S.) is the reciprocal of Ty the time between frames
Aa/S = V3 VF.S./640 WfR (10)
If the acceleration error (Aa) is required to be less than one G and if only one acceleration value is required (which makes S = 2), for a velocity of 1000 fps and a resolving power of 10
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December 1952 Journal of the SMPTE Vol. 59