Math Forum - Project of the Month, December 1996

December POM - Winner

Margaret Whitman Germantown Academy, Fort Washington, Pennsylvania

From: ruth@mathforum.org (Ruth Carver)

A range finder, often used in military (sighting for guns) and photographic equipment, is an instrument that determines the distance from a home reference point to a distant object by optical and trigonometric means. Light from the object passes directly through a fixed, semitransparent mirror so that a person can see and keep the image of the object in view. Light from the object also travels to another mirror, which reflects the image to the semitransparent mirror. The viewer sees two images superimposed on each other, and the angle of the second mirror is adjusted until the direct image and the indirect one coincide. The distance between the two mirrors is fixed by the equipment so that the angle to which the second mirror is adjusted can be determined. The distance between the object and the instrument is then calculated by dividing the distance between the mirrors by the tangent of that angle. The accuracy of the measurement is limited mainly by how well the viewer can get the images to coincide and also by the distance between the mirrors, which must be maximized within practical limits of the equipment. In more complex range finders, the mirrors may be replaced by penta prisms, which reduce problems of alignment. Telescopes may be built into the system to provide magnification, and a coincidence prism may be used to split the field of view into a top and a bottom half that the viewer must bring into coincidence.

The particular model of range finder that you spoke of is accurate within 22 yards at 400 yards has a + or - 5.5% discrepency. I got that by dividing 22 yards by 400 yards and multiplying by 100 to get a percent. The range finder that you described is about 10 inches long. A more accurate model would have to be much larger in both length and diameter, assuming that it is round. The accuracy of a range finder increases with its size. The muzzle velocity of every gun varies depending on the type of ammunition it is using and also its calibre. The smaller the calibre the greater the velocity. In order to utilize the range finder with a + or - 5.5% accuracy a hunter firing at a pole 10 yards high and wanting to hit it dead center at 5 yards would have to aim either .275 yards high or low. I got the + or - .275 yards by taking the height of the target, 5 yards and adding or subtracting to that the 5.5%. This formula would compensate for the trajectory that the bullet makes.

This was a fun problem because I got my dad to help me a bit and I don't often do that.