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Technical
Take-Offs: Where to Aim?

Nick Furse considers whether there is a good theory to cover aiming in take-off shots. The parallelogram of forces and hence the resultant is appropriate to the velocities or momentum, whereas the final resting places (used for the midpoint aiming method) are dependent on the square of the velocities (proportional to energy), given the assumptions. His conclusion is that you can hit much more into the croqueted ball.

About a year ago, when discussing where to aim the mallet in a croquet stroke I wrote “Incidentally in the case of take-offs, for which I believe the maths does work, neither method is correct.”

The two methods were:

1) ‘Half the angle’ (i.e. halving the angle between the directions of the two balls and aiming in that direction)

2) Half the distance (i.e. aiming at the midpoint of the line joining the required final positions of the two balls when they come to rest).

It was suggested by a correspondent:

“The midpoint [half the distance] method is correct, both in theory and in practice, if you allow for the factors I have mentioned.”

I believe this is not so - except in the special case where both balls are required to travel the same distance at right angles to each other (Indeed, in this special case, both the above methods work).

My results are good news for those who worry about fine take-offs (in case they don’t move or shake the croqueted ball) because they suggest that the striker can afford to aim further ‘into’ the croqueted ball than the midpoint rule would dictate. My results will also allow greater accuracy in judging where to aim for thicker take-offs.

The factors which the correspondent wished to ignore are ‘things like "pull", "mallet drag", "ball slip" and spin’.

I will also ignore such things. In addition, I will ignore the effects of the ‘Coefficient of Restitution‘, COR, and, to avoid complications with the possible effect of the mallet, I will assume that the striker’s ball is a very short distance from the croqueted ball such that it has just left the mallet and is travelling with velocity ‘v’ when it strikes the croqueted ball and, because the angle between the initial direction of the mallet swing and the ‘line of centres’ of the two balls on impact is 45° or greater (the conditions for a take-off), no double tap occurs. (The purists might consider this a close angled ‘scatter shot’).

As I understand it, the argument for believing that ‘aiming at the midpoint’ is correct is based on the following reasoning:

  1. In order for the principle of the conservation of momentum to apply:
    1. Any momentum gained by the croqueted ball at right angles to the line of aim must be exactly balanced by an equal quantity of momentum gained by the striker’s ball in the opposite direction.
    2. Since the masses of the two balls are the same, then the velocity components of the two balls at right angles to the line of aim must also be the same but in opposite directions.
  2. If a constant force slows a moving ball, the distance travelled by that ball will be proportional to the square of its initial velocity.

This last argument has led some to believe that the distances travelled by each ball at right angles to the line of aim will be the same (but in opposite directions) and thus the midpoint of the line joining the final positions of the two balls will lie exactly on the line of aim.

However, those that reason this way forget that the actual decelerating force that acts on each ball does not act in its entirety at right angles to the line of aim but rather in the opposite direction to that the ball is actually travelling. This means that only a component of that force is working at right angles to the line of aim. Furthermore these components will be different for each ball unless each ball is travelling at the same angle relative to the ‘line of aim’ (which will only happen at exactly 45°).

Variables used in the theory

In practice, in a take-off, the striker’s ball makes a narrower angle (α) with the line of aim than does the croqueted ball (90° - α), thus a smaller component of the frictional force acts on it at right angles to the line of aim than that which acts on the croqueted ball. This means that, when the balls come to rest, the striker’s ball will have travelled further at right angles to the line of aim than the croqueted ball will have done in the opposite direction. Thus the correct ‘line of aim’ is to aim ‘further into’ the croqueted ball than the midpoint of a notional line joining the desired stopping positions of each ball.

I have included the mathematics for this in an Appendix:

If we define

  1. The angle ‘α’ as the angle at which the mallet is aimed into the croqueted ball (i.e. if α = 0, the croqueted ball would not move) for the purpose of the stroke to send each balls a desired distance
  2. The angle ‘β’ as the angle which is in the direction of the ‘midpoint’ of the notional line joining the balls when they come to rest after having travelled the desired distance.

The theory then predicts that angle α is such that tan(α) = √ tan(β) or,

tan(α) = √ (croqueted ball distance) / (striker’s ball distance)

tan(α) = √ CD / SD

tan(α) = AS / SD = √ CD / SD.

Thus

AS = √ CD.SD

In practical terms, to give one example, this means that, if you wish to take-off from corner 1 to hoop 3 (about 100 feet away) and don’t want to move the croqueted ball more than about 1 foot, you can afford to aim into the croqueted ball about 10 feet to the side of hoop 3 (rather than only 1 foot as the ‘midpoint rule’ would suggest). This is because tan(β) = 1/100 and tan(α) = √  tan(β) = √ 0.01 = 0.1

This knowledge should help players avoid failing to move or shake the croqueted ball in fine take-offs without fear that the croqueted ball will roll off the lawn (unless there is an awkward slope in the corner or the grass is more worn and provides less resistance).

Nick (Furse)

Appendix

The purpose of this Appendix is to set out the theory of where to aim in a take-off stroke so that each ball travels the desired distance.

The article above explains why ‘aiming at the midpoint’ (of the notional line joining the finishing locations of the two balls) is wrong (except when each ball is required to travel the same distance).

For the purpose of this analysis I will ignore such things as "pull", "mallet drag", "ball slip" and spin”.

In addition, I will ignore the effects of the ‘Coefficient of Restitution (‘COR’) and, to avoid complications with the possible effect of the mallet, I will assume that the striker’s ball is a very short distance from the croqueted ball such that it has just left the mallet and is travelling with velocity ‘v’ in the direction of aim when it strikes the croqueted ball and, because the angle between the initial direction of the mallet swing and the ‘line of centres’ of the two balls on impact is 45° or greater (the conditions for a take-off), no double tap occurs. (The purists might consider this a close angled ‘scatter shot’ – hence describing the non-striker’s ball as the ‘croqueted ball’).

Other assumptions are:

  1. The mallet is aimed at an angle ‘α’ into the croqueted ball (i.e. if α = 0, the croqueted ball would not move).
  2. The mass of both balls is the same = m.
  3. The speed of the lawn (i.e. deceleration/ resistance due to friction, etc.) is the same in all directions.
  4. Newton’s Laws of Motion apply.
  5. ‘.’ means ‘multiplied by’.
  6.   = square root

Then, from the laws of conservation of momentum and energy, immediately after impact:

  1. The initial velocity of the croqueted ball is v.sin(α) in the direction of line of centres which is (90 - α) degrees from the line of aim.
  2. The initial velocity of the striker’s ball is v.cos(α) and its direction is at an angle α from the line of aim (but on the opposite side).

A. Conservation of Momentum

The velocities of the two balls can each be broken down into two components: one in the direction of the line of aim and one at right angles to it.

The two components in the line of aim are:

v.sin(α).sin(α) for the croqueted ball, and

v.cos(α).cos(α) for the striker’s ball.

Thus the total momentum in the direction of the line of aim is

m.v.sin2(α) + m.v.cos2(α) =  m.v.[sin2(α) + cos2(α)].

However, sin2(α) + cos2(α) = 1 (this is a trigonometric theorem). Thus m.v.[sin2(α)+ cos2(α)] = m.v.1 = m.v = the initial momentum of the striker’s ball immediately before impact in the direction of the line of aim.

The two components of velocity at right angles to the line of aim are v.sin(α).cos(α) for the croqueted ball and v.cos(α).sin(α) for the striker’s ball. These two components are in opposite directions. Thus the total momentum at right angles to the line of aim = m.v.sin(α).cos(α) – m.v.cos(α).sin(α) = 0.

Thus the principle of the conservation of momentum is preserved as the total momentum immediately before and after impact is = m.v.

Note that the line of aim goes exactly through the centre of a notional line joining the ends of the velocity vectors of the two balls and the ratio of those velocities = [v.sin(α)] / [v.cos(α)] = tan(α)

B. Conservation of Energy

Similarly, the total energy before impact = ½.m.v2. After impact the total energy = ½.m.(v.sin(α))2 (for the croqueted ball) + ½.m.(v.cos(α))2 (for the striker’s ball).

Thus total energy immediately after impact = ½.m.(v2).[sin2(α) + cos2(α)]. Since [sin2(α) + cos2(α)] = 1, total energy after impact = ½.m.v2.1 = total energy before impact.

C. Distances travelled

The distance travelled by the croqueted ball will be proportional to its velocity squared; v2.sin2(α) and the distance travelled by the striker’s ball will be proportional to its velocity squared; v2.cos2(α).

These two balls will travel at right angles to each other. If a line were drawn notionally between the points at which the two balls come to rest, the angle β from the point of impact to the midpoint of that line would be such that

tan(β) = [v2.sin2(α)] / [v2.cos2(α)] = [sin(α) / cos(α)]2 = tan2(α)

Note that tan(β) does NOT equal tan(α) except when α = β = 45° (and tan(α) and tan(β) both = 1)

Conversely, if the desired distances for both balls are known, then angle α can be estimated from the expression:

tan(α) = √ tan(β) = √ distance travelled by croqueted ball / distance travelled by striker’s ball.

Thus angle α is considerably larger than angle β (which would involve aiming at the ‘midpoint’).

For example, when playing a take-off on the West boundary parallel to hoop 2 with the intention of sending the croqueted ball to hoop 2 and the striker’s ball to corner 1 (about 4 times as far), the mallet should be aimed at the midpoint of the South boundary rather than a point immediately behind hoop 1 on the South boundary. (tan(β) = ¼ and thus α should be chosen such that tan(α) = ½).

Similarly, if you wish to take-off from corner 1 to hoop 3 (about 100 feet away) and don’t want to move the croqueted ball more than about 1 foot, you can afford to aim into the croqueted ball about 10 feet to the side of hoop 3 (rather than only 1 foot as the ‘midpoint rule’ would suggest). This is because tan(β) =1/100 and tan(α) = √ tan(β) = √ 0.01 = 0.1

This knowledge should help players avoid failing to move or shake the croqueted ball in fine take-offs without fear that the croqueted ball will roll off the lawn (unless there is an awkward slope in the corner or the grass is more worn and provides less resistance).

Nick (Furse)

Author: Nick Furse
All rights reserved © 2009


Updated 28.i.16
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