"The procedure is essentially simple," writes Graham, "but the reliability of the results will reflect the care with which certain measurements are made. I pick a sunny day for the experiment and, having arrived at the golf course with my co-operative friend and accessories, tee my ball. Then I place a tee marker precisely four feet in front of the ball and another four feet behind it to make a line that points toward the first green. My friend stations the tripod-mounted camera for a medium close-up shot on a line that intersects the ball at right angles to the tee markers. I address the ball, facing the camera. My friend photographs the complete drive from address to follow-through at the rate of 48 frames per second. The known distance between the tee markers and their position in relation to the club head scales the pictures with respect to distance. The exposure rate-the number of frames per second-of the camera provides the time dimension. (If the exposure rate is not known accurately, it can be calibrated by photographing a phonograph turntable marked with a chalk line and turning at 45 or 78 revolutions per minute.)

"The film is developed and analyzed. One can use either a film-editing device that projects an enlarged image of each frame or a set of enlarged prints of each frame, mounted serially and numbered for identification.

"The next step is to plot the position of the club head during the course of the swing. Since a point in a plane is determined by its distance from two other known points, the position of the club head can be plotted in relation to that of the two tee markers [see illustration below]. First, I draw a base line near the bottom of a sheet of graph paper ruled with rectangular co-ordinates and on it locate three equally spaced points: the tee marker P, the ball (O) and the tee marker Q. I usually space these points four inches apart, thus establishing a scale of 12 inches of club head travel per inch of graph paper.

"The location of the club head (C) with respect to that of the tee markers can be transferred to the graph by one of three methods. Proportional dividers are handy for transferring the scaled distance from P to C and from C to Q. Alternatively, the angles CPQ and CQP can be measured with a protractor and reconstructed on the graph, point C being located at the intersection of lines projected from P and Q. If no protractor is at hand, the vertical and horizontal distances between C, P and Q can be measured with a square and ruler and similarly transferred to the graph.

"Plot enough points to establish a reasonably smooth track, skipping several frames du mg slow portions of the swing. The resulting graph is of course not extremely accurate. The plane in which the club head swings, for example, is inclined to the plane of the film. The track plotted from the image therefore differs slightly from the true excursion of the club head, but the error is not large and can be ignored. By the same token, the travel of the club head from point to point is subsequently measured along straight lines, whereas the club head actually follows a curved path. Error introduced by this source can be minimized by speeding up the camera. My camera, an inexpensive one, is limited to a maximum speed of 48 frames per second, a rate that records the event adequately for the objectives of this experiment.

"The total distance traveled by the club head and its velocity and acceleration are derived from a second set of graphs prepared from the graph of club head position. On a second sheet of graph paper ruled with rectangular coordinates divide the abscissa into a series of uniform increments equal to the total number of frames occupied by the swing and note the corresponding time intervals in seconds as well as the frame numbers. The ordinate will carry two scales: club head travel in feet and club head speed in miles per hour. The scales of the ordinate should provide for a total club head travel of 36 feet and a maximum velocity of about 80 miles per hour. Graphs of convenient proportion result when the length of the ordinate representing 36 feet equals the length representing one second on the abscissa. The maximum velocity of 80 miles per hour need not occupy more than half of the ordinate scale, as shown in the accompanying graph.

"Data for plotting club head travel against time are derived by measuring the graph of club head position. Make a table of three columns, for frame number, time and distance. Beginning with the point on the graph of club head travel that shows the head addressing the ball, scale the distance to the next point and convert to equivalent feet by referring the measurement to the base line that includes P, O and Q. Measure and tabulate the remaining position points in the same way. When the table is complete, add the distance increments progressively, plot distance against time and draw a smooth curve through the points.

"The speed of the club head at any point is found from this graph by the familiar graphical method of slopes. To find the speed of the club head at about the point of impact (frame No. 43), draw a tangent LKM of arbitrary length through K. The sides MN and LN are found by referring to the scale to equal 11.2 feet and .11 second respectively. The speed of the club head at this instant is equal to the ratio 11.2/.11, or 102 feet per second. The result can be expressed in miles per hour by multiplying it by the number of seconds per hour and dividing the product by the number of feet per mile: 102 X 3,600/5,280 = 70 miles per hour. Repeat the procedure for each of the frames, tabulate the results, plot speed versus time and draw a smooth curve through the points.

"Club head acceleration can be graphed in the same way or merely computed from the graph of club head speed at frames of particular interest, such as the frame showing the moment of impact. For example, to determine the acceleration of the club head depicted by frame No. 38, draw a tangent to the graph at this point. Then, at some arbitrary point above, say at the point corresponding to a velocity of 56 miles per hour, drop a perpendicular MN from the tangent. At another arbitrary point below, say at the point corresponding to a velocity of 12 miles per hour, draw a line LN parallel to the abscissa and intersecting both the tangent and MN. Inspection of the abscissa discloses that the length LN is analogous to a time interval of .1 second. Acceleration is defined as the rate of change of velocity and is equal to the difference between the final velocity and initial velocity divided by the time interval between the two. In this example the velocity difference is 56 miles per hour minus 12 miles per hour, or, expressed in feet per second: (56 - 12) X 5,280/3,600 = 64 feet per second. The acceleration is 64/.1 = 640 feet per second per second. The acceleration of gravity (g) amounts to 32 feet per second per second. The acceleration of the club head at frame No. 38 in terms of g is accordingly 640/32, or 20 g!

"Having performed this rainy-afternoon portion of the procedure, what reward awaits the duffer? For one thing, he can see at a glance why his drives do not match those of a professional golfer. The graphs discussed so far show the performance of golf professional Dick Bull using an iron. His swing from address to follow-through required 1.17 seconds. The club head traveled 31 feet. His backswing occupied .6 second. He paused at the top about .1 second. More interesting than these figures, in my opinion, are those of the club head speed and acceleration Bull achieved: the increase in club head speed during the .1 second before impact from 15 miles per hour to an amazing 70 miles per hour, representing an acceleration of slightly over 20 g. Graphs of Bull's performance with a driver, although different in many respects from those of his irons, show exactly the same figure for speed, 70 miles per hour, and an acceleration of 22 g, a remarkably uniform performance. Similar analysis of the performance of a fairly good amateur using a driver shows precisely half the velocity of Bull's club, 35 miles per hour, and an acceleration at impact of only seven g [see lower illustration below].

"Although these methods of analyzing motion are routine in engineering circles, I am not familiar with their prior application to the game of golf. As with many procedures, they are easier to apply than to describe. I find them interesting because they clearly reveal why Bull and other professionals achieve their long drives. Duffers with movie cameras may well begin asking each other, 'How's your v and g?'"

"Interest in "ground effect" aircraft- machines that ride a few inches above the surface on a cushion of low-pressure air created by a vertical jet-led Robert W. Moffat, a graduate student at Wayne State University in Detroit, to design an inexpensive smoke tunnel with which he can make remarkably accurate aerodynamic measurements. The most novel feature of the tunnel is the use of a high-fidelity loud-speaker powered by ordinary 110-volt, 60-cycle house current to produce precisely timed pulses of smoke that, when photographed against the black background of the tunnel's interior, resemble a series of dotted lines. The shape of the lines depicts the pattern of the air flow; the spacing between the timed smoke puffs enables the experimenter to compute the velocity of the flow.

"The tunnel has seven principal sections," writes Moffat, "an air-inlet bell, the smoke system, the test nozzle, the tunnel body, the lighting system, the blower and air controls and the flow-measuring station [see top illustration at right]. Because the tunnel was designed to investigate the aerodynamics of hover craft, the apparatus was fitted with a single nozzle that directs air at right angles against the floor of the tunnel. The floor, which simulates the ground, can be fixed at any distance between two and 20 inches from the jet for investigating the behavior of air flow in the range of low altitudes within which these craft normally fly.

"To achieve the stable flow of air required for the accurate measurement of air flow in two dimensions, the apparatus was equipped with a semicircular inlet chamber, resembling half of a shallow cheese box, for smoothing the turbulence of the air in the room. The curved edge of the chamber is made of eight semicircular strips of 26-gauge sheet metal spaced 3/8 inch apart. The sheets are supported by eight slotted, streamlined struts four inches long. The sides are closed by two semicircular sheets of 1/4-inch plywood and the bottom by a straight piece of one-inch plywood with a rectangular outlet for the test nozzle, as shown in the accompanying illustration. The sides must be braced externally to prevent bowing that would disturb the flow. Air enters the sheet-metal slots through six layers of fly screening. This construction limits the velocity of air flowing from the room into the screening to about 15 per cent of the jet velocity at the point where air enters the tunnel.

"The smoke system is supported on the front face of the air-inlet unit. It consists of a forced-draft burner that admits tobacco smoke to a pulsing apparatus from which puffs of smoke are delivered by Tygon tubing to a set of seven 1/8-inch copper tubes mounted in a radial array inside the inlet bell, as illustrated. The smoke generator consists of a straight metal tube two inches wide and 16 inches long closed at one end by a rubber stopper drilled for a cigarette or cigar holder and supplied with air at the other end by an electric hair dryer. Pressure inside the smoke generator is adjusted by a controlled leak–three slots that can be adjusted in size by a sliding rubber sleeve.

Bibliography

THE AERODYNAMICS OF POWERED FLIGHT. Robert L. Carroll. John Wiley & Sons, Inc., 1960.

FLOW VISUALIZATION IN TWO AND THREE DIMENSIONAL FLOW FIELDS BY USE OF SMOKE FILAMENTS. A. M. Lippisch in ASME Symposium on Flow Visualization, Paper No. 2, pages 1-7; November, 1960.

THE SCIENTIFIC AMERICAN BOOK OF PROJECTS FOR THE AMATEUR SCIENTIST. C. L. Stong. Simon and Schuster, Inc., 1960.

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