The serve and return are perhaps the two most important shots in a game of pickleball. On the serving side, a fast and well-placed serve can be played for an outright winner, or it can put the receiving team at a significant disadvantage by forcing them deeper behind the baseline or by making them hit poor returns. On the receiving side, being able to predict and anticipate the path (or trajectory) of a pickleball serve and judge its speed will allow you to better position yourself for an awesome return shot. In this article, we will apply the equations of motion to a pickleball serve to determine how fast you can serve a pickleball.
This article is third in the series describing “Pickleball Dynamics“. The reader is advised to read the previous article, “Pickleball Equations of Motion” to gain a better understanding of the equations used in this article.
The fastest tennis serve clocked in at over 163 mph (263 km/hr) by Sam Groth at the Busan Open in South Korea on May 9, 2012 . However, it is well known that a pickleball serve is much slower, with a groundstroke clocking in at a maximum speed of only 40 mph (64 km/hr). Why so slow? Could the large difference in speed be attributed to differences in racquet and ball design, weight, materials, spin, etc.? As we will see in this article, the actual reason that a pickleball serve is slower than a tennis serve is much more fundamental, which can be explained if we examine the equations of motion.
Pickleball Equations of Motion
Figure 1 shows the dimensions of a pickleball court. For the time being, let’s assume that the ball is served at the center of the baseline and lands at the center of the opposite baseline. The center of the ball must reach a height of about 36” (3’) to clear the net, which is located 22’ from either baseline. By regulation, the pickleball must be contacted below the height of the server’s waist, and the point of contact on the racquet must be below the server’s wrist. For a typical underhanded serve, the point of contact is below the height of the server’s knee, let’s say about 18” or 1.5’, which is one-half the height of the net.
The equations of motion of the ball in the horizontal and vertical direction are:
x(t) = x0 + vxt Equation 1
y(t) = y0 + vyt + ½ gt2 Equation 2
Where vx and vy are the components of the ball velocity in the horizontal and vertical directions, respectively, x0 and y0 are the initial positions of the pickleball (-22 ft and 1.5 ft respectively), and g is the acceleration due to gravity (-32.17 ft/sec2 or -9.81 m/s2). Our goal is to calculate the fastest possible serve velocity while clearing the top of the net and landing in bounds at the opposite baseline. For the time being, we will neglect aerodynamic drag, aerodynamic lift (spin), and wind.
From trigonometry, we can express the x- and y- direction components of velocity in terms of the initial velocity (v0) by:
vx = v0 cos θ
vy = v0 sin θ
Where θ is the angle of contact with respect to the horizontal that the pickleball paddle makes with the ball. Substituting,
x(t) = x0 + v0 (cos θ) t Equation 3
y(t) = y0 + v0 (sin θ) t + ½ gt2 Equation 4
The (Theoretical) Fastest Pickleball Serve
We can solve Equations 3 and 4 by knowing that the fastest serve will have a trajectory that just barely clears the top of the net. That is, a higher serve will take more time to arc over the net and must therefore have a lower velocity (we can prove this with our equations of motion). If a pickleball is about 3” in diameter, the center of the ball must reach a maximum height (or apex) of 37.5” (3.125’) to just clear the net. Furthermore, at the apex the vertical velocity component vy must equal zero, since the ball’s direction changes from rising in the +y direction to falling in the -y direction.
The solution of the equations of motion yields the trajectory curve shown in Figure 2. Voila(!), the initial velocity (v0) of the fastest pickleball serve is almost 40 mph at a contact angle of 10.5 degrees!
These results demonstrate that a pickleball serve must be slow because the geometry of the serve dictates it. Since regulations require that the contact point of a pickleball serve be below the waist (and by extension, below the height of the net), the pickleball must rise against gravity to its apex at the top of the net, then gravity takes over, allowing it to fall within the court before reaching the baseline. No other combination of velocity and contact angle can satisfy our constraints of serving the ball at a height of 1.5 ft, clearing the net at 3.125 ft, and landing in bounds within the far baseline at 44 ft. Hitting the ball faster than 40 mph will result in the ball over-shooting the far baseline and reducing the contact angle below 10.5 degrees will cause the ball to hit the net.
You may now be wondering about the velocity of a short serve that barely clears the kitchen line. Solving Equations 3 and 4 where the final horizontal position x(t) is just outside of the opposite kitchen line at 7’ beyond the net yields the trajectory shown in Figure 3. Here, the contact angle (θ) of the serve must be about 25 degrees and the initial serve velocity v0 is about 23 mph. A slower serve velocity or a higher or lower contact angle will not allow the ball to clear the net.
These results also suggest that in the absence of aerodynamic drag, wind, or lift (spin), a pickleball player is limited to hit the ball on a serve with a minimum velocity of about 23 mph and a maximum velocity of about 40 mph. Serves that fall out of this range will either hit the net or sail beyond the baseline. Pickleball players must be cognizant of the fact that when they serve, they are not aiming and shooting for the landing point like you would shoot a target in archery, but you are controlling the ball velocity and serve angle to essentially drop the ball onto the desired landing point. So in a sense, serving in pickleball is like shooting a basketball.
The Importance of Aerodynamics
One thing you should notice with the short serve is that the apex (or highest point) of the serve is about 4.5’ and occurs about 9’ before reaching the net. However, experience suggests that good pickleball players can make low serves that just clear the net and land just outside the kitchen line. Similarly, we know from experience that good pickleball players can serve the ball faster than 40 mph while keeping the ball in bounds.
These apparent discrepancies are due to the fact that these results are “ideal” because they neglect aerodynamic drag, wind, and lift (spin). As we will see in “Can Topspin Enable a Faster Serve?“, your ability to apply spin to the ball will allow you to hit the ball at higher velocities while keeping the ball in-bounds. Aerodynamics makes the pickleball trajectory calculations more difficult, but they also make the game of pickleball much more challenging, interesting, and unpredictable.
