Aerodynamic drag is always present in pickleball, and is dependent on numerous variables, including the speed, mass and surface condition of the ball. In this article, we develop a mathematical model of aerodynamic drag and use it to predict the maximum speed of a pickleball serve.
In our article, “Pickleball Equations of Motion“, we introduced the equations of motion of a pickleball and analyzed its trajectory from the instant that it is hit by the paddle until it lands within the court on the other side of the net. In “How Fast is a Pickleball Serve?“, we demonstrated that a pickleball needs to reach a velocity of only 40 mph to follow the trajectory of the fastest possible pickleball serve. This result does not seem to make much sense, as we know from experience that a great deal of effort goes into hitting a fast serve that must at some point exceed 40 mph.
The reason for this lies in the fact that our previous analysis assumed that the flight of the pickleball was affected only by the force of gravity. In reality, the pickleball encounters aerodynamic drag (and lift) as it travels through the air. This article sheds light on the origins of aerodynamic drag and how it affects the trajectory of a pickleball during a serve.
Elements of Aerodynamic Drag
Aerodynamic drag always acts in a direction that is opposite that of the velocity. If we examine the free body diagram of the pickleball (Figure 1), the horizontal component of drag always acts to reduce the horizontal velocity of the ball. The direction of the vertical component of drag depends on whether the ball is rising or falling. In Figure 1a, with the ball rising, the vertical component of drag acts to slow the balls rise. In Figure 1b, with the ball falling, the vertical component of drag acts to keep the ball afloat.
An interesting, and somewhat related topic involves the concept of “terminal velocity”. Many people think that an object’s terminal velocity is the fastest that it can travel through the air, which is not entirely correct. The terminal velocity of an object is established by dropping the object from a high height then measuring its steady state velocity. As shown in Figure 2, the velocity vector (v) points downward in the direction of gravity, and the drag force points upward, opposing gravity. When the steady state terminal velocity is reached, the drag force exactly balances the gravity force. The ball therefore cannot exceed the terminal velocity when it is falling under the influence of gravity only. That is not to say that a pickleball cannot travel faster than its terminal velocity. If you could fire a pickleball from a cannon, it could certainly travel faster than its terminal velocity. We will not normally need to deal with the idea of terminal velocity in pickleball.
The Drag Equation
In the technical literature, it is known that if an object travels slowly, the amount of aerodynamic drag is linearly proportional to its velocity. Linear (Stokes) drag problems are relatively easy to solve mathematically. Unfortunately, for objects such as pickleballs, tennis balls, baseballs, footballs, golf balls, etc. the drag force is proportional to the velocity squared – that is, doubling the velocity will quadruple the drag force acting on the object. This drag model, known as quadratic drag (or Newton drag), is mathematically complex and does not have an exact analytical solution. The quadratic drag force, FDrag acting on a moving object is defined by:
Let’s take a look at each of the components that make up the drag force.
- CD is the drag coefficient, which is dependent on the smoothness and velocity of the ball in the flow field (air). While typical values of CD can fall in the range from 0.2 to 1.0, NASA Glen Research Center has tested and analyzed balls in wind tunnels and found that a smooth ball will have a CD of 0.5, and a baseball (with seams) will have a CD of 0.3. This result sounds counter-intuitive – why does a smooth ball have more drag than a rough ball? The answer lies in the fact that a smooth ball produces less turbulence as it travels through the air, causing the air molecules to stick to the smooth surface and impede its travel through the air. For our purposes, we will assume that the CD of a pickleball is about 0.4. We can analyze this topic in depth in a future article.
- ρ (the Greek letter “rho”) is the air density. A ball played at sea level in Pompano Beach, FL will have much more drag than a ball played at altitude in Albuquerque, NM. Matches played at altitude will therefore have faster moving balls than those played at sea level due to the lower drag force.
- A is the cross-sectional area of the ball, which is a constant.
Drag-Enhanced Equations of Motion
Our approach to solving this problem will be to add the drag force into our equations of motion by dividing the force by the mass of the ball, yielding a drag acceleration that always acts in a direction opposite that of the velocity vector. Since we must divide the drag force by the mass of the ball, a heavier ball will have a lower acceleration rate due to drag than a lighter ball. The drag-updated equations of motion are shown in Equations 1 and 2.
These equations do not have an exact analytical solution because the acceleration components must constantly change in both magnitude and direction as the ball travels through the air. Fortunately, we can solve this problem numerically assuming that the velocity is piece-wise continuous over a short time step, allowing us to solve the problem in a recursive manner.
Fastest Pickleball Serve with Drag
Now that we have developed the drag mathematics, we are going to apply Equations 1 and 2 to predict the fastest serve with aerodynamic drag. Using the input data in Table 1, we our goal was to determine the initial velocity (v0) and the contact angle (θ) required to match the trajectory of the fastest pickleball serve where aerodynamic drag is neglected (see “How Fast is a Pickleball Serve?” Figure 2 shows a comparison of the fastest serves with and without drag.
Table 1. Input Data
In this analysis (shown by the red and green curves), the center of the ball clears the net with and without aerodynamic drag by about the same amount – 1.50”. If the ball diameter is 2.90”, then the surface of the ball comes to within 0.05” from the top of the net. Both balls land in-bounds at the baseline (+22.00 feet).
What Does This Mean?
These results show that it is almost possible to match the trajectory of the fastest pickleball serve without aerodynamic drag by serving the ball at an initial velocity of about 54 mph. A pickleball player must therefore supply about 35% more velocity (and force) to their serve in order to match the trajectory of the fastest serve without drag.
These analyses also show that over its course of travel, the ball velocity reduces from its initial velocity of 54 mph the instant it leaves the racquet to a velocity of about 31 mph the instant it strikes the ground on the opposite side of the court. About 43% of the ball velocity is therefore lost to aerodynamic drag!
The elapsed time of travel (or time of flight) from the instant the racquet strikes the ball until it lands on the ground is the same for both cases — about 0.77 sec! This might seem non-intuitive for some readers, as we might expect that the presence of drag will reduce the ball velocity causing the time of flight to be longer. However, the important idea here is that just as long as we can supply greater force to the pickleball to match the trajectory of the fastest serve, the average velocity will be the same in a vacuum or in the presence of aerodynamic drag, wind, or lift (spin). So, a serve with a 54 mph initial velocity that follows the trajectory shown in Figure 2 must have an average velocity of about 40 mph. The good news is that in the absence of wind and spin, no serve can reach the opposite baseline in less than 0.77 sec.
