Detecting the Enemy. 1 Ship to Ship combat

Transcription

1 Detecting the Enemy 1 Ship to Ship combat Let s take a trip back in time to World War I England. Imagine yourself a sailor on a warship traveling in the seas around England, hunting enemy warships. One of your mates is in the crow s nest using a telescope to look for enemy ships and suddenly he spots one. He is too far away to see if it is friend or foe. Your ship wants to get closer to check it out and engage in battle if it is an enemy. Though the telescope can give you information about what angle to steer the ship, it doesn t tell you how far away the other ship is. This can be useful information and the process of finding it out is called optical range finding (in present days we would use radar, but radar had not yet been invented). Below is a picture of the situation. Points A and B are on either end of the ship, a distance a apart. At those points, sailors are looking at the target through telescopes and they report the angle they must turn 1

2 through to see the second ship (this angle is measured as if the sailors were originally facing the back of the ship). These angles are called bearing angles and it requires two of them to compute the distance between the two ships. We will call these two angles θ 1 and θ 2, and they are measured from the line segment BZ. The distance between A and B can t be too large because they are constrained by the ship, but we want them far enough apart that we get two different bearing angles. The point C is the position of the unknown ship. We want to find out the distance the second ship is from ours, knowing only the bearing angles. We will denote this distance as r and we will assume r = AC. We assume that r is very much larger than a (for example a is measured in yards and r is measured in miles). Our goal here is to determine a formula for r and to understand how sensitive our computation for r is to errors in the bearing angles and how r depends on a. 1. Explain why ACB = θ 1 θ 2 2. Solve for r. 3. In general for small ψ, sin ψ ψ (provided φ is measured in radians). Use this to show that to a good approximation r a sin(θ 2) = a sin(θ 2) θ 1 θ 2 φ where φ = θ 1 θ 2 (consider trigonometry rules for non-right triangles). Thus we have a formula for r in terms of a, θ 1 and θ 2, all variables whose values we know. That s just what we wanted. Now we know the exact position of the other ship. Or do we? 4. We also want to know how errors in measuring the bearing angles affect our measurement of r. After all, the measurements we get for the bearing angles will not be perfect. In addition, we want to know how r depends on a. There is not much chance that our measurement for a will be off by very much because we can measure it directly and it s easy to be fairly accurate when measuring short distances. Remember that if one variable depends on another and you want to know how a change in one effects the other, you can differentiate. What derivative do you want to compute to see how r depends on the difference of the bearing angles? Compute this. Note that φ also depends on a, so the error estimate is clearer if we eliminate φ using φ = a sin θ 2 /r. Show that we get r2 dr a sin(θ 2 ) dφ. 2

3 5. Suppose a = 100 yards, r is measured to be two miles and θ 2 is radians (all of these values are approximations). What is the coefficient of dφ in feet? in miles? Argue that we want θ 1 θ 2 to be very small. If θ 1 θ 2 =.01 radians, what is dr in miles? (To help you picture this, how many degrees is.01 radians?) Do you think θ 1 and θ 2 can be measured precisely enough to guarantee that dr is small? What effect does it have on our error estimate to have r 2 on top? Why do we want a to be as large as possible (though it will always be much smaller than r 1 and r 2 )? 6. We don t have to use derivatives to understand the dependence of r on φ. We could instead make a spreadsheet that shows values of r a sin(θ 2 )/φ for different values of φ = θ 1 θ 2. Try this for values of φ starting at.01 and increasing by.01 each step. Use a = 100 yards and θ 2 = 70. How sensitive is r to changes in φ? You can also graph r as a function of φ. Your graph should have a steep derivative (i.e. slope). Clearly optical range finding is very inaccurate and a more accurate way of measuring the position of enemy ships was needed. 2 Air Combat 2.1 Radar is Born At first during World War I, airplanes were used for reconnaissance purposes, but near the end of the war they started being used as bombers, though at first their payload was things like bricks and grenades. During the break between World War I and World War II, countries started experimenting with more sophisticated uses of airplanes. Britain established a small committee of scientists to study the problem of fending off bombers and the chair of this committee was one Henry Tizard. During the First World War, the only way Britain had of detecting bombers was by sight. Someone in a fighter plane would have to be flying in the vicinity of the bomber, see it, and try to shoot it down with a machine gun. Many ideas were brought to Tizard s group. A man by the name of Robert Watson-Watt presented the idea of a death ray to the committee. The death ray would be on the ground and aimed at enemy airplanes. The idea was that this ray would burn the pilots to death. The ray would be some strange concoction of ultraviolet rays, x-rays, and radio waves. It was actually decided to try to make such a device and Watson-Watt s assistant, Arnold Wilkins, was asked to carry out experiments testing the efficacy of such a death ray. The scientists found out that their death ray would raise the temperature of a person standing 600 meters away by 2 degrees Celsius in 10 minutes. To do this the ray needed to produce many thousands of kilowatts. The idea of a death ray was given up, however, this experiment led Wilkins to consider using radio waves to detect enemy aircraft. Radio waves would be sent out concentrically from a transmitter and if an object was hit a wave would be reflected back to the transmitter. One problem with radio waves is that the reflected signal has only of the power of the original transmitted wave, so the original wave must be very powerful in order to detect a reflected signal. To transmit a powerful enough wave, the transmitter must be very large. 3

4 Yet Watson-Watt threw his weight behind Wilkins idea and experiments began. Watson- Watt is commonly accepted as the inventor of radar. Since the transmitted waves had to be so powerful, huge radar units were made. Radio waves would be transmitted concentrically from either end of the radar unit. Why do we need two points of transmission? These first radar systems were too huge to put on an airplane, or even a ship. Instead, large radar units were mounted on tall poles and placed along the coast to look for enemy airplanes (radar units were made small enough to fit on ships in 1938, but these were too large to fit on aircraft; radar units on aircraft didn t come til the end of the war). A controller on the ground would have to radio pilots as to the position of the enemy aircraft. Since the radio waves are sent out concentrically from the radar system, the distance of the enemy aircraft can be determined, but we have to do some more work to find out its exact position. 4

5 Suppose A and B are points, from which radio waves are transmitted, at either side of the radar unit, and suppose C is the enemy target. We want to find out exactly where C is. We know two distances r 1 = AC and r 2 = BC. To determine the exact location of the enemy we need to know the bearing angle θ. We assume that r 1 and r 2 are much larger than a = AB (for example a may be measured in yards and r 1 and r 2 in miles). 1. Mark X on the line CB so that CX = AC. Show that to a very close approximation So cos θ r 2 r 1. a ( ) r2 r 1 θ arccos a and we have all the information about the enemy target s position that we need. 2. Deduce that if we make an error ɛ in measuring r 2 r 1 ; that is, ɛ = (r 2 r 1 ), then the resulting error in the calculation of cos θ, that is cos θ, is ɛ/a. Thus the error in calculating cos θ is inversely proportional to a. So, again, we want a to be as large as possible (though it will still be much smaller than r 1 and r 2 ). 3. Let s = r 2 r 1. We measure r 1 and r 2 using radar, but there will be some error in this. We saw how easy it was to show relationships using derivatives, rather than computing function values. Show, using derivatives, that the error in calculating θ with respect to s is also inversely proportional to a. (What derivative do we want to compute to 5

6 see how changes in θ depend on changes in s?) So, again, we want a to be as large as possible. 4. Suppose a is 100 yards again. Argue that dθ ds Give a convincing argument that θ can be measured fairly accurately. As you can see, radar is much more accurate than optical range finding. 2.2 Directing Interception Once people knew how to use radar to determine the position of an enemy target, they had to figure out how to use this information to direct a fighter or ship to intercept the enemy. For the remainder of this section we assume the enemy target is a bomber and we are trying to intercept the bomber with a fighter, but the same techniques could be used to direct ships. 1. Consider a simplified problem in 2 dimensions. We can tell from our radar readings that a bomber is seemingly flying along the x-axis at a constant speed of u, so its position is given by (ut, 0). A fighter starts at (x 0, y 0 ) with constant speed v in a direction θ (the angle is measured in degrees below a line parallel to the x-axis going through (x 0, y 0 ) and we will call it the flight angle) in an attempt to intercept the bomber. The fighter s position can be described as (x 0 + vt cos(θ), y 0 + vt sin(θ)). If u = 20 and v = 30 and (x 0, y 0 ) = (2, 5) (we are measuring everything in miles per hour), how long will it take a fighter to intercept the bomber and at what angle should the pilot fly? If the fighter flies at a greater flight angle than the one you computed (i.e. more negative), what will happen? Because of radar, the fighter can intercept the bomber without actually seeing it. 6

7 Computing the flight angle for the fighter, even in this simple situation, requires a lot of computation. 2. Describe two more situations in which the fighter would intercept the bomber. If the fighter starts in quadrants 2 or 3 what will it have to do to intercept the bomber? If the fighter is in quadrants 1 or 4, what will it have to do to intercept the bomber? Also describe two situations in which the fighter will not intercept the bomber. Support your claims, but don t find the angles and speeds as we did in the first problem. Base your argument on common sense. 3. Suppose the bomber started out at the origin and flew at an angle 30 degrees from the x-axis with velocity 100 and the fighter was at (3, 6) with angle θ with velocity 150. At what angle must the fighter fly to intercept the bomber and how long will it take? (Hint: Rotate the picture so that the bomber is flying along the x-axis.) Again, what if the fighter flies at a larger angle (i.e. more negative)? The British carried out many experiments to see how best to use radar to intercept an enemy bomber. The first experiments were done with the bomber flying in a straight line, with its initial position determined by radar. In a few weeks radio directed fighters were able to intercept the bombers almost every time. But once the bombers were allowed to change course and altitude, the success rate was greatly reduced. Evasive flying on the bomber s part meant that the calculations involved in producing intercept courses had to be repeated over and over again and that the information on which these calculations were based was probably out of date by the time the pilots got it. Remember that there were actual people on the ground doing these exact computations and radioing up courses to the pilots. We saw in the problems above how time consuming computing the flight angle can be Tizard s Angle Tizard came up with a simple solution to this problem based on the fact that fighters fly faster than bombers. If the bomber is at point A traveling along the line AD, and the fighter is at point B, then the controller should direct the fighter along the line BC which intersects AD at point C. C should be chosen so that AB is the base of an isosceles triangle ACB. 7

8 1. Explain why if neither fighter nor bomber change course, the fighter will always arrive at point C first, assuming it flies faster than the bomber. 2. By drawing accurate diagrams, investigate what will happen if the bomber continues in a straight line course but the controller alters the fighter s course at regular intervals in accordance with Tizard s rule. 3. Explain why, even if the bomber changes course from time to time, a controller altering a fighter s course at regular intervals in accordance with Tizard s rule will still achieve interception. You will have to assume that the controller alters the fighter s course at substantially shorter intervals than the bomber changes course. However a bomber taking constant evasive action against an opponent will probably not make much progress towards the target. Assume that the bomber does not backtrack, just to make the situation easier to draw. Rather than computing the flight angle the way we did above, we use information from the radar to compute Tizards angle, which can be done using a protractor. This is much less time consuming than the original method for computing the flight angle. We can compute the flight angle for the fighter using Tizard s idea and the information we get from the radar. There was one major problem left and that is that radar systems had to be large and so had to be mounted on tall towers built into the ground. Then a controller would have to transmit information to either a plane or a ship. The key to the size of the radar system was the wavelength of the the waves it produced. A radio transmitter cannot produce high power radio waves of a wavelength much greater than its size. Remember, we need to produce high powered radio waves since the signal that bounces back to the radar set has power only of the original transmitted wave. Scientists managed to produce a system that would work on 1.5 meters wavelength. The original towers worked with a wavelength of 13 meters, so this was a sufficient reduction- sufficient enough that radar systems could be put on ships. It was some time before radar units could be made small enough to be put on airplanesthat didn t happen until the the war was nearing its end. 3 Summary In this project we first learned how to determine the location of an enemy through optical range finding; that is, given the bearing angles we computed the distance of the enemy. We talked about this method in terms of ship-to-ship warfare. For airplanes, optical range finding doesn t work very well because the targets are moving so fast and with optical range finding, the pilot would have to be able to see the enemy aircraft. At the time when ships were using 8

9 optical range finding, planes were using visual location. Once radar was discovered, it took the place of optical range finding and visual identification. Radar could find the distance of the enemy target, and do so much more accurately than with optical range finding, and then a person could compute the bearing angle and work out an interception course and radio these instructions to a plane or ship. When the enemy s position changed, this would have to be done over again, but these computations were lengthy. Tizard realized that if you chose a course that would be the leg of an isosceles triangle (with the enemy s flight path being the other leg), then you could intercept the enemy. This was a big breakthrough, as measuring Tizard s angle only required a protractor, and, in fact, controllers got good enough that they could guess Tizards angle bye eye. If the enemy changed course, the new Tizard angle would be computed. Radar units were originally too large to be put on a ship or plane and instead were put atop tall towers. Controllers would collect the data from the radar and compute the flight angle, and radio that information to planes or ships. Eventually scientists were able to make a radar set that could be used on a ship, but it was not til the end of World War II that they could build ones small to be put on planes. 9