In October's column, I noted that several technical papers on the engineering, science and technology related to tennis were presented at the Engineering of Sport Conference at the University of California, Davis. One of the most interesting and practical papers was presented by Prof. Howard Brody titled, "Where to hit on the racket for maximum power and reduced errors" [1].

The premise of the work is that most unforced errors are due to errors related to ball depth (either the ball hits the net or is long over the baseline). Hence strategies which correct ball depth problems improve your game.

You may already know that there are three "sweet spots" on a tennis racquet: the center of percussion, the node and the maximum power point. If you hit a ball at the center of percussion you minimize the shock; if you hit the ball at the node it minimizes vibration; and if you hit it at the maximum power point your ball has its fastest rebound. While the center of percussion and node are fixed points determined strictly by the geometry and characteristics of the frame, the maximum power point is determined by those parameters as well as the incoming ball speed and stroke. Consequently, maximum power point is different for each player and shot.

How do you determine power in a racquet? Researchers launch balls at a suspended racquet and determine the ratio of the ball's speed off the racquet to the ball's speed onto the racquet (V_rebound/V_incident) called the apparent coefficient of restitution (ACOR).

ACOR is maximized near the center of mass or balance point of a racquet and is reduced as a ball is hit away from this point. This is because when the ball strikes the racquet away from the balance point some of the energy rotates the racquet, so there is less energy to generate the ball's rebound speed. When the ball strikes the center of mass, the racquet does not rotate.

If you were able to move the racquet in a straight line, so that all parts of the racquet moved at the same speed, then the maximum ACOR and the maximum power location would be identical. However when you swing the racquet, the racquet makes an arc, and the racquet tip has a faster speed than the throat. This moves the maximum power location higher up the racquet head.

Brody couples this with previous work ("How would a physicist design a tennis racket?" [2]), which demonstrated that as the ratio of ball incident speed to racquet head speed increases, the maximum power point moves down from the tip towards the throat.

Bottom line: Brody advises: "Therefore, it is recommended that when playing on fast courts (such as grass), the ball should be hit a little closer to the throat. When playing on slow courts (such as clay), the ball should be hit a little closer to the racket's tip. It is also clear from this argument that a serve should be hit closer to the tip and a volley, closer to the throat of the racket."

Aside from the obvious advantage that you will attain the desired ball speed with less effort on the racquet swing, Brody demonstrates through two examples that hitting the ball at the maximum power position will reduce long shots (over the baseline) and short shots (which typically give your opponent an easier shot to return). Assume that the spin, height and angle are constant in both examples and appropriate to produce a good ball trajectory. Hit a ball at 70 mph at the peak power point such that the ball lands deep inside the court (close to the baseline). If you slightly miss-hit it on the racquet (either higher or lower on the racquet), the ball velocity will be slightly slower in either case but the ball will still land inside the opponent's court. However if you swing the racquet to produce a 70 mph ball deep into the court but hit it below the peak point and slightly miss-hit, then a ball hit closer to the tip will produce a faster ball and go out.

While the maximum power position can only be found by trial-and-error, Brody provides a general rule of thumb: "the greater the racket head velocity is, relative to the incoming ball velocity, the farther out on the racket the ball should be hit."

However, if you hit the ball and if you hit it just above or below this point there is slightly less power, and these will also land inside the court. Overall this should reduce unforced errors due to hitting the ball long.

I wish you all a very happy and healthy New Year. Until Next Month ... Jani