The Physics Behind the Perfect Baseball Swing
Jaeheon Lee
Watching baseball's greatest hitters like Shohei Ohtani send home run after home run deep into the stands makes you wonder: how can a hitter send the ball that far and still control where it goes?
In an MLB broadcast, two things are always mentioned when evaluating a hitter: exit velocity and launch angle. Exit velocity measures how fast the ball leaves the bat, while launch angle describes the direction of the ball after contact. Although a higher exit velocity generally allows the ball to travel farther, the angle of contact plays a major role in determining whether it will clear the fence. Because of this, players and coaches try to optimize launch angle to increase the chance of hitting home runs.
Simple projectile motion provides a useful starting point for studying this problem. In the idealized case, assuming no air resistance, a constant initial speed, and equal launch and landing heights, the maximum horizontal range occurs at a launch angle of 45°. Because a baseball is struck above ground level, the equal-height assumption does not hold exactly in practice. I therefore use the 45° result only as a theoretical baseline and incorporate a realistic initial contact height in the Euler model.
However, this does not match what happens in real baseball. MLB Statcast data shows that many home runs are hit at launch angles between roughly 25° and 35°, much lower than the theoretical 45°. This suggests that the simple projectile model does not fully represent the motion of a baseball. One major missing factor is air resistance, which acts against the ball’s motion and reduces its speed throughout the flight.
Because air resistance changes continuously with the speed of the ball, the motion becomes difficult to solve using the standard projectile equations. I therefore used Euler’s method, which estimates the ball’s motion over many small time intervals by repeatedly updating its velocity and position. This allowed me to model the trajectory while including drag.
Using this numerical model, I calculated the horizontal range for launch angles between 10° and 80°. The distance increased until the angle reached about 41°, where the maximum range was approximately 119 m, and then began to decrease. This shows that including air resistance lowers the predicted optimal angle from 45° to about 41°. A flatter trajectory keeps more of the initial velocity in the horizontal direction and also reduces the time the ball spends losing energy to drag.
Even so, 41° is still higher than the launch angles commonly seen in real baseball. To compare the model with actual performance, I analyzed batted-ball data from Francisco Lindor. Lindor is a switch hitter with whom I have always forged a personal connection, as I have also worked to develop my own ability as a switch hitter. To keep the comparison as consistent as possible, I filtered the data for balls hit between 105.0 mph and 106.9 mph, close to the 106 mph initial velocity used in my model. This allowed launch angle to remain the main variable being compared.
Exit Velocity (mph) | Angle (º) | Range (ft) | Range (m) |
106 | 27 | 404 | 123 |
106 | 43 | 356 | 109 |
106 | -8 | 14 | 4.27 |
106.1 | 23 | 394 | 120 |
106.1 | 3 | 82 | 25.0 |
Lindor’s data showed a clear nonlinear relationship between launch angle and horizontal distance. The range increased toward a peak and then decreased more sharply afterward, creating an asymmetric pattern. Because of this, I used cubic regression rather than a simple quadratic model. The regression produced an R² value of 0.948, indicating a strong fit. Using differentiation to find the local maximum, the optimal launch angle was estimated to be approximately 32.6°.
The results therefore show a clear progression. The ideal projectile model predicts an optimal angle of 45°. Including air resistance lowers it to approximately 41°, while Lindor’s real data gives an optimal angle of about 32.6°. One possible reason for the remaining difference is the Magnus effect caused by ball spin. Backspin can create lift and keep the ball in the air longer at lower launch angles, allowing flatter trajectories to travel farther than the drag-only model predicts.
Overall, the analysis shows that the theoretical 45° launch angle is only a starting point. Incorporating air resistance into the Euler model lowers the predicted optimal angle to approximately 41°, bringing the theoretical result closer to real-world behavior. However, the Statcast regression suggests an optimal launch angle of about 32.6°, revealing a remaining discrepancy between the model and observed data.
This difference suggests that factors not included in the model, such as spin, atmospheric conditions, spray direction, and variations in aerodynamic drag, may also play an important role. These factors would need to be incorporated and tested in a more complete model of the baseball’s trajectory.
Overall Comments:
The main direction of this article is strong, but you need to make the comparison between theory, simulation, and MLB data more rigorous. First, state all assumptions and parameters behind both the 45° and 41° results so the numerical model can actually be reproduced. Second, be careful saying launch angle is the main remaining variable in the Lindor dataset because spin, atmospheric conditions, spray direction, and other factors still affect distance even when exit velocity is controlled. Most importantly, justify the cubic regression more carefully by reporting your sample size, showing the distribution of data around the predicted optimum, and avoiding excessive precision in the 32.6° estimate. You also need to try to feel more like a news article, integrating the whole theory or story with some actual recent events.
){kind=small}
){kind=small}
){kind=small}
){kind=small}
