PHYS-1500 PHYSICAL MODELING Class 11: A Compound Bow FALL 2005 NAME ____________ANSWERS___________ When an arrow is launched by a bow, the arrow acquires kinetic energy equal to the work done on it by the bow (ignoring frictional effects). For bows of classical design, such as that used by Robin Hood, the force exerted by the bow is approximately proportional to its displacement from its equilibrium position. Therefore, such bows work much like a simple spring. However, for a compound bow, a relatively recent development in the field of archery, the relationship between force and displacement is far from a straight line. In this problem, you will investigate some of the differences between a standard ("recurve") bow and a compound bow. The Excel file "bows.xls" contains tables of forces exerted by recurve and compound bows for over 300 different displacements from equilibrium. These data have been graphed to illustrate the very non-linear nature of the compound bow. Even though neither of the graphs is a straight line, it is possible to calculate the work done by the bows as functions of the distance they have been pulled back. Graphs of work vs. displacement have been set up, but they will be blank when the spreadsheet is first opened. They will be filled in automatically when you calculate the work done by the bows. You can open the Excel file by double clicking on the icon below. Work is the area under the graph of F vs. x, so the area from x = 0 to any selected value of x gives the work done when the bow is released from that value of x. To calculate this area, you can calculate the area of each of the vertical strips between adjacent data points. These can be calculated with the help of the formula for the area of a trapezoid. For points 1 and 2 this formula is, dW = (x2 – x1)(F2 + F1)/2. That can be interpreted as the width of the base of the strip times the average value of the height. Enter the formula that will calculate the area of these strips in the column marked dW. For the recurve bow, enter the formula for the strip from C8 to C9 in cell E9. The formula is, =(C9 – C8)*(D8 + D9)/2. Then use the "fill down" command to copy the formula into cells E10 through E309. To get the area from x = 0 to any value of x, the areas of these strips must be added up (integrated), and the running sum listed. Column G contains the values of x again, for graphing purposes, and work can be listed in column H, which is labeled "W(J)". The first entry in the work column is zero, in cell H8, and the rest of the cells are blank. Cell H9 should contain the sum of the entry in H8 plus the work done in the first strip. Cell H10 should contain the sum of the value in H9 plus the work done in the second strip, etc. This can be accomplished by entering in cell H9, the formula, =E9+H8 and then copying it into cells H10 through H309 with the "Fill Down" command. When you have completed this, the graph of work vs. displacement for the conventional bow will appear. Repeat the entire process for the compound bow data. The column numbers are different, but the procedure is the same. Also note that there are more data for the compound bow, so when "fill down" is used, the formulae must be copied to cells numbered up to 359. On the paper that you turn in: 1. Then, record the work done by each bow when x is pulled to about 45 cm (0.45 m), the point at which the force exerted by the compound bow reaches a local minimum. Wrec = _______58.0 J________ units Wcom = _____64.3 J__________ units 2. Which bow does more work when pulled 45 cm? (Circle the correct choice.) RECURVE COMPOUND 3. Which bow requires the greater force while being aimed? (Circle the correct choice.) RECURVE COMPOUND 4. Calculate the speed of an arrow fired from each bow if it released from x = 45 cm (0.45 m), the point at which the force exerted by the compound bow reaches a local minimum. The mass of the arrow is given in the spreadsheet. It is m = 0.036 kg = 36 g. vrec = _____56.8 m/s__________ units vcom = ______59.8 m/s________ units 5. Do you see why compound bows have become popular? Yes. They require less force while aiming, and they produce greater arrow speed.