"Although the club face's direction at separation doesn't determine the direction of the ball's flight, I didn't exactly say that the ball leaves the club face in the direction of the force at the ball's greatest deformation. I said that that direction would be a fairly good approximation of the ball's direction (much better than the face's direction at separation, anyway).Got it?

The ball is influenced by the club only between the time of first contact (let's call that tc) and the time of separation (ts). The velocity vector at separation (Vs) will equal the integral of its acceleration vector over time, from tc to ts. The acceleration vector equals the vector of net force on the ball (F) divided by the ball's mass (m). F is actually a function of time, expressed as F(t). It changes in both magnitude and direction during the impact interval. It is zero at tc and ts, climbs to a maximum magnitude of several thousand pounds at the time of maximum ball compression, and must average about 1400 pounds to accelerate the ball from 0 to 150 mph within half a millisecond (the approximate difference between ts and tc). Mathematically, then,

Vs = Integral from tc to ts of F(t)/m dt.

The direction of F(t) will be dominated by the club face's normal vector at time t. There might be a smaller contribution from frictional shear forces if the club head's velocity vector doesn't align with the club face's normal vector at any given time. This will, of course, be true if the club has a nonzero loft, but we're not interested in the vertical components of the forces and velocities here. Let's define a right-handed coordinate system with x in the direction that our assumed right-handed golfer would face at set-up, y in the direction of his target, and z in the vertical direction. From this point on, we will ignore the z direction, treating F(t) and Vs as two-dimensional vectors with x and y components only (as though the club had zero loft).

Let's first consider Homer Kelley's "angled hinging", during which the club head's x-y velocity vector will always align with the club face's x-y normal vector. Assuming the club head doesn't twist from an off-center hit during impact, there will theoretically be no shearing forces on the ball (imparting no side-spin, contrary to Kelley's assertions about angled hinging), so F(t) will always align with the face's normal vector (technically, the dynamics of ball compression and decompression might affect the direction of F slightly, but we will ignore any such potential complications).

If the swing is designed to have the club face's normal vector aligned with the y direction (i.e., pointing at the target) at separation, then the normal vector will have a zero x component at ts, but it will have a positive x component for all previous times (since the club face is continually closing as its normal vector follows the club head's path). As a result, F(t), which aligns with the normal vector, and which is zero at tc and ts, will have a positive x component for all times in between. Therefore, from the integral above, Vs is guaranteed to have a positive x component. In other words, the ball will be pushed somewhat to the right of the target.

Similarly, if the club face is squared to the target line at tc, then its normal vector, hence F(t), will have a negative component for all subsequent times, guaranteeing that Vs will have a negative x component. In other words, the ball will be pulled somewhat to the left of the target.

For the ball to fly directly at the target, Vs must have a zero x component. The only way that can happen is for F(t), and therefore the club face's normal vector, to have negative x components during the later portions of the impact interval to offset the positive x components from the earlier portions. Therefore, the club face must be square to the target line about midway between tc and ts. The exact time (let's call it tm) would be very difficult to calculate, but a good approximation would be the time of maximum compression. That would also make sense, intuitively, since that is when the club will exert most of its influence on the ball.

The situation is a little more complicated with "horizontal hinging", during which the club face is square to the club head's path only when it is square to the target line. Before that time, the club face will be slightly open to the target line (creating a normal force with a positive x component) and to the club head path (adding a clockwise shearing force to the ball). After that time, the club face will be slightly closed to the target line (creating a normal force with a negative x component) and to the club head path (adding a counter-clockwise shearing force). For this type of swing, it is even more important to have the club face square to the target line about midway between tc and ts. Prior to the club face becoming square to the target line (and to the club head path), F(t) will still have a positive x component, but now, because of the shearing contribution, F(t) will be directed slightly to the left of the ball's center. That will induce a clockwise torque on the ball, giving it a slicing side-spin. If the club face doesn't become square until ts, the ball will be guaranteed to have a positive x component for Vs and a non-zero clockwise angular velocity. In other words, the ball will be both pushed and sliced somewhat to the right of the target. Similarly, if the club face is squared at tc, the normal vector, and hence F(t), will have a negative x component for all subsequent times. In addition, because of the shearing contribution, F(t) will be directed slightly to the right of the ball's center, inducing a counter-clockwise torque throughout the impact interval. The ball will therefore be guaranteed to have a negative x component for Vs and a non-zero counter-clockwise side-spin. In other words, the ball will be both pulled and hooked to the left of the target.

For the ball to fly directly at the target, with little or no side-spin, the horizontal hinging must be performed in a way that squares the club face to the target line (and the club head path) about midway between tc and ts, say at tm. Between tc and tm, F(t) will have a positive x component and will cause a clockwise torque (and hence, angular acceleration) on the ball. Between tm and ts, F(t) will have a negative x component and will cause a counter-clockwise torque (and hence, angular acceleration) on the ball. A judicious choice of tm will cause the x components of F(t) and the opposing directions of angular acceleration to integrate to nearly zero (it might not be possible to exactly zero both simultaneously, however), giving a ball flight straight at the target with little or no side-spin. The exact value of tm would again be very difficult to calculate, but a good approximation would again be the time of maximum compression, which is again when the club exerts most of its influence on the ball.

Now that's some good stuff. Pretty obvious really, but good stuff none-the-less. See its o.k. to say 'we have the science to prove it' but unless you publish for peer review then its just your opinion. I'd love to see a lot more of this kind of stuff here.