When A Mallet Strikes A Ball
By Stan Hall
The main purpose of the tests described below was to determine the duration of the contact between a mallet and a ball and also how far contact time varies with the initial velocity of the ball, being longer for small velocities. The initial velocity is not easy to measure but it can be accurately calculated if the lawn characteristics are known. Hence the first tests are concerned with measuring the coefficients of friction.
Figure 1 shows a velocity-time graph of the ball. The slope of the graph is the acceleration of the ball and the area under the graph is the distance travelled. The graph consists of three phases: -
The coefficient of sliding friction was measured directly. Three balls were embedded in a sheet of polystyrene, which was then pulled along by a weight acting through a string and pulley arrangement. The string was attached close to ground level to minimise any overturning which would result in an uneven reaction on the three balls. A spring balance measured the force at the edge of the ball assembly. For a given weight (i.e. pulling force) the ball assembly was given a push to overcome static friction. If the assembly then moved no further the weight was slightly increased. Eventually the assembly then moved along at uniform speed after an initial push. The spring balance reading then gave the coefficient of sliding friction. For a fairly fast lawn (normal, cut to 6mm) the coefficient was 0.48. For a fairly heavy lawn, the coefficient was 0.53.
Once ms is known, the coefficient of rolling friction, mr, can be found by measuring the total distance, s, and the total time of travel, t, for several single ball strokes (see equation 8). For a normal lawn mr is about 0.065. For grass cut to 4mm, it is 0.055, and for a very heavy lawn a value of 0.1 was obtained.
Various relationships between time and distance can now be found from the geometry of Figure 1.
The time interval during sliding is
(1) and the distance travelled is
The time interval during rolling is
and the distance travelled is
Ignoring very small distances, s1, the total distance travelled is
If we take ms/ mr as being approximately 7.5, then
The sliding phase thus occupies a fraction of the total travel given by
or about 10%. Also from equation 6 we have
mr = 0.065
metres per second, or
feet per second.
For the purpose of calculating mr the total distance travelled may be taken as the area of the triangle EOG in Figure 1. The error in neglecting triangle ECB is about 1.6%.
Theoretical calculation2 show that the shape of the velocity-time curve during the time interval t1 is as shown in Figure 2 with the time scale greatly exaggerated. The area under this curve gives the distance travelled as approximately
For this purpose, a few strands of copper wire were wound around the ball. At right-angles to this, a strip of aluminium foil was fixed to the ball. Foil was also taped to the end faces of the mallet. The two metal surfaces were connected to an electronic timer so that when the surfaces were in contact the timer was operational. As an alternative to the timer, a cathode ray oscilloscope (CRO) was used in some tests. This has the advantage of showing the detail during the contact period. Preliminary tests were carried out on both Dawson balls and Jaques balls. No significant different could be noticed.
For long strokes, the ball was struck so that it would normally travel the
required distance, but its progress was shortly stopped by a cushion to prevent
damage to the wiring and foil. An error of say 10% in the distance does not
significantly affect the contact time. The results are given in Table 1.
Numerous attempts were made to keep the ball in contact with the mallet for a longer period as in a "push stroke". This proved difficult. One attempt was particularly successful, producing a total contact time of 3ms. However, the graph on the screen of the CRO showed that in reality there were three separate contacts of 1ms each with a gap of 1/10ms between them. The distinction between the contacts could not be detected by the unaided senses. A double tap became audible with an interval of 6ms between contacts.
A hammer stroke was tested with the shaft of the mallet inclined at about 70 degrees to the vertical. The contact time was in excess of 7ms (the CRO only registered up to this time). The long contact time was probably due to confinement of the ball between the mallet and the ground.
The tests on croquet strokes must be regarded as somewhat preliminary. The number of variables precluded a thorough investigation at this stage. However, the tests clarified the general trend of behaviour.
For these tests, two circuits were used, one to monitor the contact between the mallet and the striker's ball (call this ball 1), and another for contact between ball 1 and ball 2 (the croqueted ball). Graphs of both circuits were displayed on the screen of the CRO.
The first test examined drive strokes in which the ball is struck as for a
single ball stroke with the shaft approximately vertical. Both balls moved in
the same direction, i.e. the angle of split = 0. Different tests sent the balls
different distances, the ratio R being kept constant at 0.3. Similar tests were
then made an angle of split of 45 degrees. The results are given in Table 2.
In all these tests the contact time between the mallet and ball 1 exceeded the contact time between ball 1 and ball 2. The time at which ball 2 moved away from ball 1 was fairly constant at about 1.8ms. The mallet remained in contact with ball 1 for a further period of 1.1 ms on average.
The effect of varying R, the ratio of the striker's ball travel to croqueted ball travel, was briefly investigated. The previous series of tests was repeated with R = 0.7 instead of 0.3. This required a different grip. Not all players use the same grip to produce this change of R, so whether the results would vary for different players is not yet known. In the present tests, the results were almost identical to those for R = 0.3. On average, the time of contact between the balls was 1.7ms and the mallet remained in contact with ball 1 for a further period of 1.1ms.
The effect of increasing R to 1.0 (full roll) was tried. This had little effect on tc, the time of departure of ball 2, but the time of contact between the mallet and ball 1 increased dramatically to a value in excess of 7ms.
Several stop shots were made. The results did not differ noticeably from those of the previous croquet strokes. As before, the shorter strokes resulted in longer contact times than did the harder strokes.
Finally take-off strokes were tried. In these strokes the striker's ball is not impeded by the croqueted ball. The times of contact between the mallet and ball agreed well with the time for single ball strokes. Unlike other croquet strokes contact between the mallet and striker's ball ended before contact between the two balls. This agrees with the idea that prolonged contact of mallet and ball in most croquet strokes is due to the interference of the croqueted ball. This effect is almost entirely absent in the case of the take-off. Slightly more surprising was the fact that contact between the balls continued well after the mallet contact had ended, but this is in line with the short travel of the croqueted ball which calls for a longer contact time.
The purpose of these tests was to determine how close a ball needs to be to the leg of a hoop for a crush to occur. As for the croquet strokes two circuits were set up, one to monitor contact between the mallet and the ball and a second for contact between the ball and hoop. Two series were carried out which will be called Type 1 and Type 2.
In Type 1 the ball struck the leg a glancing blow and was free to depart thereafter (Figure 4). This refers to a situation where the ball could run the hoop without touching it but for some reason slight contact is made. It also covers the case of a slightly angled hoop where minor contact is necessary. It is noted that the distance travelled by the ball before contact is greater than the gap, since the ball does not approach the hoop directly.
Results are shown in Table 3. The degree of contact is expressed in terms of
the angular deviation of the ball upon contact. The type of stroke is designated
Strokes were full follow-through unless indicated by a "jab" or "gm" (grounded mallet).
For this type of stroke it seems that a crush will not occur if the gap is 1/16th inch or more. There were two exceptions - a hard stroke combined with either a considerable deviation or a loose hoop.
In Type 2 tests the ball was driven directly towards the centre of the leg. This concerns very angled hoops where the ball entirely comes to rest in the hoop or where it nearly comes to rest but later runs the hoop. In these situations the mallet may catch up with the ball after separation has taken place. The ball then oscillates between the mallet and the hoop. This is recorded in Table 4 as a multiple contact. A true crush was not observed although presumably this could occur after several oscillations. In any case the stroke is illegal. Results are given in Table 4.
* multiple contacts between mallet and ball - time given is the time gap to the first re-contact
For single ball strokes the contact time is almost constant at 1ms except for very short distances in which case the time is longer. The distance travelled during contact varies with the total travel of the ball - about 0.15 in, for a very long stroke down to 0.06in for a short stroke.
In croquet strokes the time of contact for the croqueted ball (contact between balls) appeared to be governed only by the distance travelled by that ball - generally about 1.8ms but longer for short distances. The contact time between mallet and striker's ball exceeded the contact time between the balls for all except take-off strokes, the excess time being about 1.1ms. In a take-off the two contacts appeared to be governed independently by the two travel distances, being shorter for the striker's ball which in turn went further than the other.
There was a strong indication that contact between the mallet and the striker's ball is increased when free movement of the ball is hampered in any way - by the presence of the front ball in a croquet stroke, or by the ground if the mallet is inclined forward. When both of these effects are present the increase in contact time can be very considerable.
In the case of crush strokes, a crush rarely occurs from a glancing blow if the initial gap is 1/16th inch or more. When the ball entirely or nearly comes to rest after striking the hoop, a crush (or multiple tap) is almost inevitable if the gap is less than 1 inch. For greater distances much depends on the nature of the stroke. Usually the duration of the stroke will now be long enough for the referee to be able to observe the stroke.
g = gravitational acceleration
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