The Physics of a Rising Fastball

It appears as though the legend of the rising fastball has actually been debunked. Baseball analysts now talk about “apparent rise” or “what rise looks like.”  Even “MythBusters” has described the rising fastball as “busted.”

That said, Alan Nathan pointed out, “A fastball could rise in principle. It could actually rise if you could get enough spin on it.”  The point here is to investigate just exactly how much spin is needed. Unfortunately, it will require a bit of physics – so, try to stay awake for the next few paragraphs, OK?

To simplify matters, let’s just think about the two forces that act vertically on a pitch in flight. The first is pretty easy to understand – gravity pulls the ball downward toward Earth.  This is the weight of the ball which is established by MLB.

Rule 1.09 – “The ball shall be a sphere formed by yarn wound around a small core of cork, rubber or similar material, covered with two strips of white horsehide or cowhide, tightly stitched together. It shall weigh not less than five nor more than 5¼ ounces avoirdupois and measure not less than nine nor more than 9¼ inches in circumference.”

For the rest of this article, we’ll take the middle road and use five and one eighth ounces which is 0.32 pounds for the downward force on a pitched ball. Now the upward force – this is more complicated.

The upward force on the ball is called the “Magnus force” or the “lift,” but that’s just a name. The Magnus force is caused by the air through which the ball is traveling. It is due to the interaction between the spin of the ball and said air. I’ve written in detail about the underlying cause of the force on a spinning ball previously, so I’ll spare you those details here.

The Magnus force is usually written as where C_L is the “lift coefficient,” r is the density of the air, A is the cross-sectional area of the ball, and v is the speed of the ball. These quantities all seem pretty straight forward with the exception of the lift coefficient which is anything but easy to understand.

The lift coefficient is extremely complicated to actually figure out. It depends upon the speed of the ball and the backspin in a pretty complex way (see “The effect of spin on the flight of a baseball” by Alan Nathan). There is still some debate as to the exact relationship, but for those that care, here is the equation I used for the lift coefficient,

where w is the backspin in revolutions per minute (rpm) and v is the speed of the ball in miles per hour (mph).

To summarize, we can compare the size of the Magnus force which depends upon the speed and backspin on the ball to the weight of the ball. After all is said and done, if the upward Magnus force is greater than the downward weight of the ball, the ball will in fact rise. Otherwise, it will fall.  So, the critical condition is when the two forces are equal.

When you set the two forces equal and do a bunch of “mathy” stuff, you get the equation below describing the relationship between the speed of the ball and the backspin of the ball when the forces balance,

where again, w is the backspin in rpm and v is the speed of the ball in mph. In the graph below, the relationship at the critical point is given by the dark blue curve. If the combination of backspin and velocity is above the curve in the blue shaded area, the ball will actually rise. If the combination is below the curve, the ball will fall.

The graph has a couple interesting and pretty sensible features. When the speed drops too low, the required backspin gets impossibly large. In a similar way, as the spin rate gets small, the necessary velocity grows way too big. These features are consistent with the basic physics of the Magnus force which requires both speed and backspin to provide an upward force.

Now that all the pesky physics and math details are out of the way, we can finally have a bit of baseball fun. Let’s study a few fastballs and see how close they come to actually being able to rise. Of course, the ones we need to examine must have high velocity and high backspin. The pitch with the highest backspin is generally the four-seam fastball.  Searching the PITCHf/x database for 2016 for four-seamers gives the table below listing the top 10 pitchers in terms of the number of pitches above 100 mph.

Numero uno is no surprise – Aroldis Chapman, who hurls over half his pitches above the century mark.  In fact, Chapman threw 25 pitches over 104 mph!  Mauricio Cabrera fired 52 between 102 and 103 mph.  To see how close these guys are to throwing a truly rising fastball, let’s look at the following four scenarios.

Plotting them on the backspin vs. velocity graph will give an indication of how close these fire-ballers are to making a fastball actually move upward.  The color on the left of the table above corresponds to the data point on the graph.

Using Chapman’s average speed and backspin, you can find the lift coefficient and then the Magnus force.  I got 0.28 pounds for the Magnus force compared to the 0.32 pounds which is the weight of ball.  In other words, Chapman is surprisingly close to being able to throw the mythical rising fastball.

To be fair, I am using the starting speed of his pitches and we need to remember pitches at this speed pitches slow down by eight to nine mph during their flight to the plate. So, the Magnus force I calculated is the maximum which occurs when the ball is released. The lift will drop about ten percent on the way to home.

So, what would Chapman have to add to his already superhuman fastball to make it rise? Looking at the horizontal dashed line in the graph it appears that with the same spin (2360rpm) Aroldis would need to throw the ball at 113 mph. If instead he can maintain the speed, the vertical dashed line shows he could spin the ball at a rate of 3100 rpm.

There are other possibilities that would include lesser increases in both velocity and backspin. However, it would seem that these are beyond human mechanical abilities. Nonetheless, as a friend of mine is prone to say, it is still nice to think about.