## Determining the Ball Speed of Serves and Groundstrokes Using ACOR

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One of the most important measurements of a racquet is its Apparent Coefficient of Restitution, or ACOR for short. ACOR is a number that describes the ball bounce speed off a hand-held racquet, and ultimately, the return ball speed of your hit. ACOR is a measure of the "bounceability" of the entire racquet, not just of the strings. [Link available to USRSA members only.]

As explained in Chapter 11 of *The Physics and Technology of Tennis,* once you know the ACOR of the desired hitting location on a racquet, you can use that information to determine how fast you can hit any shot, given any incoming ball speed and any racquet swing speed at the impact point.

The formula to determine ball speeds is:

v_{out} = v_{in}·ACOR + (1 + ACOR)v_{racquet}

### Typical real-life values for slider variables:

- ACOR: 0.45 at center of strings, 0.1 at tip, 0.50 at throat.
- After the bounce, incoming ball speed is usually about one half speed it left opponent's racquet.
- Maximum racquet speed will be about 100 mph.

### Try These Settings:

- Set incoming ball speed to 0.0 mph. This simulates a serve.
- Now set ACOR to 1.0 (no energy loss). Notice that maximum possible service speed in this universe is 2 times the racquet speed.
- Set incoming speed. Notice when ACOR is 1.0, maximum possible groundstroke speed is 2 times racquet speed plus the incoming ball speed.
- Set ACOR to 0.0. No matter the incoming ball speed or racquet speed, the ball outgoing ball speed will be the same as the racquet's.
- Set ACOR to 0.45 and incoming ball speed to 0.0 mph. Now move the racquet speed slider to how fast you must swing to achieve a given service speed. For example, with ACOR = 0.45, the racquet must swing about 83 mph at the impact point.

### Interesting observations:

- Leave incoming ball speed to 0.0 mph to simulate a serve.
- For a serve, if the ACOR was a maximum of 1.0 (impossible), the maximum possible velocity is twice the racquet speed.
- With an ACOR, the maximum possible groundstroke speed is twice the racquet speed plus the incoming ball speed.
- If ACOR were 0.0, no matter what the ball's incoming speed, the ball speed after the collision is the same as the racquet speed after the collision. In other words, the ball sticks to the racquet.
- If you set ACOR at 0.45 (a typical value for the sweetspot of a racquet) and set initial ball speed to 0.0 mph, then you can move the racquet speed slider to see how fast you must swing to achieve a given service speed. For example, with an ACOR of 0.45, the racquet must swing about 83 mph at the impact

For more information about ACOR, see *The Physics and Technology of Tennis*.