The invention of the bow and arrow may rank in social impact with the invention of the art of kindling and the invention of the wheel. It must have been in prehistoric times that the first missile was launched with a bow. We do not know where and when. This may be the reason that for many students the system of bow and arrow is interesting (besides their interest for projectiles in general).
The laboratory course for undergraduate students in physics in their first year at the University of Amsterdam ends with a 'free' experiment. 'Free' means that the students work at an experiment of their own choice, and that the experiment is open-ended. Recently one of our students, who is a competition archer, proposed to measure the velocity of arrows under different circumstances. The method he proposed uses the voltage that is induced by the magnetized point of an arrow, when the arrow flies through two coils at a fixed distance. This method appeared to be very accurate and gave a number of interesting results.
For calculations on the system of bow and arrow, models have to be made. In the 1930s Hickman, Klopsteg and Nagler not only performed experiments on different designs of the bow, but they also developed models. As part of modelling simplifying assumptions had to be made in order to obtain a solution in closed form or to approximate the solution of the governing equations numerically in an acceptable amount of computing time. Because of these simplifications only bows with specific features could be described. In Section 4 we discuss some of these models in more detail.
The velocity of the arrows was determined by measuring the time of flight between two coils 2 metres apart. The first coil was placed at some distance from the bow, varying from 0.5 to 3 m. The coils used had nearly 400 windings of 0.2 mm copper wire on wooden formers with an aperture of 16.5 cm diameter. For an experienced archer it is no problem to shoot the arrows through both of the coils. To improve the experimental conditions we installed a heavy rack to which the bow and the release were attached. This release is a mechanical device which clamps the string with a small loop of cord, and which looses the arrow in a very reproducible way when a button is depressed.
Fig.1: Electronic circuit with 3 opamps of type 741, forming two amplifiers and a Schmitt trigger. The variable resistor of 1 MOhm serves to make the ratio of the resistors of the first opamp exactly equal to obtain a high common mode rejection ratio.
In each experimental situation a series of 12 shots was made, from which the average and the standard deviation were calculated. The accuracy mentioned between brackets is always the standard deviation of the set of measurements.
When shooting from hand the arrow is placed under a small clamp on the grip, the clicker. When the arrow is drawn it is glides under this clamp until the becomes free from it, and a little click is heard: the arrow is at the clicker point. This is the moment the arrow has reached its final draw length and has to be released. For shooting from the rack the mechanical release was mounted at a fixed distance relative to the grip, so we did not have to use this clicker.
The induction voltages were amplified by a differential amplifier, built with two opamps of type 741, as indicated in Figure 1. An opamp Schmitt trigger, adjusted at a small positive level and zero level, gave the edges necessary for starting and stopping the timer. To check the signal a transient recorder was connected to the output of the amplifier. An example of these signals is shown in Figure 2.
Fig.2: The form of the induction voltage and the Schmitt trigger levels.
The time dependence of the voltage induced in the coil depends on the place where the arrow flies through the coil. At the moment that the induction voltage goes through zero with a very sharp edge, the point of the arrow must be very close to the median plane of the coil. Since the same applies to both coils, the distance is defined very precisely. The point of the arrow was magnetized by attaching it for some time to the pole of a permanent magnet. For shooting from hand the bows were equipped with stabilizers. All measurements were performed indoors.
The force-draw curves were determined by hanging the bow on a hook and loading it with different masses on a scale. The masses had been measured very precisely (better than 0.1 %). The distance of draw was measured with a steel measuring staff with a millimetre division, whose accuracy is better than 0.25 %.
Other methods for measuring the velocity are photography with a high speed camera, determining flight distance and the ballistic pendulum. Nowadays, video recording is also possibility, but none of the methods can attain the accuracy of our method. Maybe the method used for measuring muzzle velocities of bullets, where the bullet passes through two 'planes' of laser bundles may equal our precision.