An early application of some of the techniques used in this class were fire control systems on naval ships in World War.  Although digital computers were not used for such systems until after the war, ships used analog computers and firing tables generated by digital computers to calculate how to hit distant enemy ships (sometimes over the horizon) during the war.  These systems had to be very sophisticated to account for the role of the ship, the movement of both ships, wind resistance, and other environmental factors.  To see how root finding would be useful, consider the following (substantial) simplification of the problem.

Assume both boats are stationary.  Ignore environmental factors except gravity.  Let y_0 be the height of the gun barrel above sea level, y be the height of the target above sea level, x be the distance from the boat to the target, and |v| be the magnitude of the barrel velocity (speed).  We need to find theta, the angle to aim our gun, to hit the other boat.  Let v_x = cos(theta), the x component of the initial velocity, and similarly v_y =  sin(theta).  We know our solution must satisfy

v_x t = x

where t is the time our shell is in the air, given as the solution to the formula

(gt^2)/2 + v_y t + y_0 - y = 0

giving us a root finding problem (with an undesirable root, which we have to watch out for as in our project)..

Also notice that the speed of all computations are critical, as in reality everything would be moving, so the computations need to be made before the state changes.  Fortunately US ships had access to precomputed tables from early digital computers built for this exact purpose (the word “computer” is used a little liberally in this sense, calculator may be more appropriate).

Round off error must have also been a very large concern, as the distances that ships were firing at required incredible precision.  Consider a second back of the envelop calculation.  Suppose both ships are again stationary and we are shooting at something 5 miles away, about 8000 meters.  For simplicity take y = y_0 = 0, g =10m/s/s.   Then v_x * v_y = 40,000, so at 45 degrees our velocity would be 200*(2^.5).  Suppose our velocity were off low by ((2^.5)/10)%, making our x and y  off by one one hundredth of a percent.  This would cause our shot to be about 20 meters short, which could be the difference between hitting and missing.

Sources:

http://en.wikipedia.org/wiki/Fire_control_computer

http://en.wikipedia.org/wiki/History_of_computer_hardware

http://en.wikipedia.org/wiki/ENIAC

http://www.navweaps.com/index_tech/tech-052.htm