## 17 September 2011

### Coefficient of Restitution

Let's think about the model of the appearance of the collision of the ball and head before and after the impact by using the coefficient of restitution .
This time let's think the model of lofted head hit the ball horizontally. It is an easy example, so-called diagonal collision .

As shown in fig
Head speed before the impact : V
Head speed after the impact : V'
Ball speed : v
Ball weight : m
Launch angle : θ
The momentum is preserved before and after the collision.
MV = mvcosθ + MV’x …①
We think only about horizontal direction (x direction element) of the coefficient of restitution .
e = - (V’x-vcosθ)/V …② From ①、②. Ball speed just after the impact is
v = M(1+e)V/((m+M)cosθ)　…③
And Head speed just after the impact is V’x = (M-em)V/(m+M)…④

There is movement of a perpendicular method (direction of y) because this is an oblique impact, too the ball is upward, and the head moves downward. However, it is possible to move only a little because the head is restrained with the shaft.
To analyze this effect, just replace ①to
0 = mvsinθ - MV’y …①’
Then become V’y = (mvsinθ)/M …⑤

If you think about not the oblique impact but a simple linear impact, you just take out cosθfrom ①②③.

Well, let's try to calculate v. (It is easy to do by Excel. )
Ball speed v becomes 58.1m/s
in the case of
V=40m/s, m=45.5g, M=192g, e=0.75, and θ=13deg and head speed V'x just after the impact becomes 26.6m/s.

You understand that the head has decelerated unexpectedly (40→26.6m/s) though it doesn't understand too much with the unassisted eye. It is natural that the head decelerated because the ball speed became 58.1m/s from 0. 