As you know tennis courts are made out of a wide variety of materials. Courts are characterized as "fast" or "slow" based on the materials and condition of the surface. As the ball hits the court it rolls or skids across the surface. If the ball skids on the court, we say that the court is fast; if it rolls, we say that the court is slow.
You know what that means to your game: grass is fast, clay is slow. On clay courts you have a little extra time to reach the ball. If you've played on the grass court you know you have less time to reach the ball -- you need to prepare earlier. You also know that you're lunging and bending a lot more since the ball doesn't rebound as high on grass.
Aside from your experience, physics tells us that there is a way to quantify court speed. The two parameters you need to understand are the coefficient of friction and the coefficient of restitution.
Coefficient of Friction
We discussed friction in an earlier section. To review, when two objects slide across one another they both exert a frictional force against one another. These forces are always tangent to the surfaces. A tennis ball and its interaction with the court is an example of this. The frictional force is opposite the direction that the ball is traveling.
The science of physics gives us the following equation: f = mN for objects that slide against one another; where f, the frictional force is equal to N the normal (upward force that the surface exerts on the ball) multiplied by m, the coefficient of friction. m is not a constant; it will vary with the ball and surface type. The more friction there is between the ball and the court the slower the ball will move after the bounce. Balls that skid on the other hand do not generate as much friction and subsequently do not slow down as much after the bounce. So, the COF tells us how fast (or slow) a ball will reach you in the horizontal direction. The higher the COF is for a court the slower the ball will be after the bounce.
Coefficient of Restitution
For ball/court interaction, the COR is a ratio of the vertical velocity after the bounce to the vertical velocity before the bounce. The COR indicates to us how high the ball will bounce.
Look at these two trajectories. Let's say that both balls have identical velocities coming off the court. We placed little circles on the graph to mark fractions of a second. Each color on each graph designates the period of time. (Red is .1 seconds, Blue .2 seconds, etc.). The x-axis (horizontal) represents distance from the bounce. If you are standing 7 feet from the point that the ball bounced, you can see from the graph that the ball bouncing off court 1 is going reach you sooner than the ball from court 2. Part of what has occurred is the horizontal velocity, the velocity component in the direction that the ball travels, is faster for the ball off of court 1 than for court 2.

Remember from our section on velocity, that velocity has both a magnitude (speed) and direction.
In trigonometry we learn that the square root of the sum of the squares of sides of a triangle is equal to the hypotenuse. We can apply this same rule to the horizontal and vertical components of velocity: V= (Vx2 + Vy2)1/2; where Vx is the horizontal distance and Vy is the vertical component. So COF deals with Vx and COR deals with Vy. (When we captured this footage one thing we did was make sure that the ball did not veer off towards either sideline. We wanted to make sure that our calculations were in the x or horizontal direction and the y or vertical direction and not the z or lateral direction which would have meant including a Vz term as well.)