Curiosity Lab / The Math of Ordinary Everyday Things / Measuring UW Sculptures with a Tennis Ball
Campus curiosity + Math 124

Measuring UW Sculptures with a Tennis Ball

A visual story about height, velocity, acceleration, derivatives, and the campus art we walk past every day.

I am always looking for ways to make calculus memorable. UW has a beautiful campus, and I do not think we use it nearly enough as a classroom.

I almost always have a tennis ball nearby—I was a college tennis player—and calculus is full of balls, tomatoes, and water balloons moving up and down. So I wondered: could I use a tennis ball to estimate the height of a campus sculpture?

The goal is not perfect measurement. The goal is to connect an actual throw, a height graph, a velocity graph, and a real campus object worth noticing.

Dr. Loveless’s Double It and Square It Rule

A shortcut I noticed while throwing tennis balls at campus sculptures.

My highly official tennis-ball rule, ha
Time the toss. Double it. Square it.
For this throw, \(T=2.83\) seconds from throw to catch. So: \(2T=5.66\), and \((2T)^2 \approx 32.0\text{ feet above my hand}\).
I threw and caught the ball at about 6 feet high, so the sculpture estimate is about \(32+6 \approx 38\) feet. That lines up nicely with the historical note: about a 14-foot bronze figure on a 24-foot pedestal.
I threw the ball several times until one toss looked close to the top. This is not really about surveying accuracy; it is about illustrating calculus, learning a little campus history, and having some fun.
As far as I can tell, I have not seen this shortcut named elsewhere, so I am claiming it for myself. It is also one of the very rare examples where feet are nicer than meters: using \(32\text{ ft/sec}^2\) for gravity makes the rule come out in feet as “double it and square it.” The metric system wins almost every other day, but not this one.

Concept 1: The Height Graph and the Velocity Graph

The ball moves straight up and down. The parabola is not the path of the ball; it is the graph of height as time passes.

Choose a campus sculpture

T = -1.00
Actual toss video — George Washington
Throw-to-catch 0.00 sec
Height parabola
h(t)
Velocity slope
v(t)

Concept 2: George Washington, Campus Art, and a False History

The toss gives us a reason to look more closely at an object many of us pass without really seeing.

Sculpture profile

George Washington

Bronze figure by Lorado Taft, dedicated at UW during the 1909 Alaska-Yukon-Pacific Exposition.

Artist Lorado Taft
Dedicated June 14, 1909
Bronze figure about 14 ft
Pedestal 24 ft
  • Pennies helped build George. The Rainier Chapter of the DAR raised money for four years, including pennies from Washington schoolchildren, no more than five cents apiece.
  • The funding was very public. The DAR raised $6,000, and the State of Washington contributed another $8,000.
  • Taft was not happy with the original placement. He thought Washington should be approached gradually, not stationed casually at the fair entrance.
  • George spent years “in the mud.” After the fair, the temporary wooden pedestal rotted; in 1920 the statue was moved to ground level on railroad ties.
  • The pedestal finally arrived 30 years later. The WPA built the 24-foot sandstone pedestal in 1938; it was dedicated in 1939, exactly the height Taft had wanted.

To learn more about the statue, the A-Y-P Exposition, Lorado Taft, and UW public art, check out the links below.

UW Libraries A-Y-P history UW Magazine mud story Lorado Taft lecture brochure UW public art
Dr. Loveless false history — absolutely not true

I tell students that George Washington founded UW, helped build the first building, and on his last day on campus they unveiled this statue for him. They named a cafe “Bye George” to say goodbye, but in a rainstorm the “e” fell off, and ever since it has been called “By George.”

I have told this story to many students and visitors to UW. I always say: if you believe any of that, then you need to go back to your high school history teacher to complain. UW was founded in 1861. George Washington died in 1799. And, no, the cafe was never called “Bye George.” Please understand that my tour of UW sculptures is not 100% historically accurate, ha.

The mud years UW Magazine tells the wonderfully undignified middle chapter: Taft’s temporary wooden pedestal decayed, George was put at ground level, and Seattle weather often turned the area muddy. A 1935 Seattle Times editorial complained that the statue had been shown “no more respect than would be paid a mass of junk.” Read the UW Magazine story →
Student life and pranks Before the permanent pedestal, students liked seeing George up close: he served as a meeting spot for dates and library rendezvous. UW Magazine also mentions pranks, including a stovepipe hat and orange sash for Junior Prom, an engineers’ surveying instrument, and later even a Bart Simpson mask.
Contested legacy The statue also belongs to a harder conversation. George Washington enslaved people, and during the 2020 protests after George Floyd’s murder, UW student activists, including UW BLM and the Black Student Union, called for the statue’s removal. The Stranger reported a month-long protest-art installation at the statue, with banner making, chalk art, poetry, and performance. Read the 2020 protest coverage →

Concept 3: What if we take air resistance seriously?

The double-and-square rule is wonderfully good for the real George Washington toss. These four pull-downs let us test how far that rule can reasonably go for a tennis ball.

1. Interactive graph: no drag vs. tennis-ball drag

This graph compares two balls with the same measured throw-to-catch time: a no-air-resistance ball and a tennis ball with quadratic drag. The green dashed graph ignores air resistance; the blue graph includes tennis-ball drag.

Start with the actual George Washington toss: \(T_{catch}=2.83\). The no-drag model gives \((2T)^2\approx32.04\) ft above my hand, while this tennis-ball drag model gives about 32.01 ft. That is only about 0.25 inches lower.
T = 2.83
This graph starts at the actual sculpture-toss value, \(T_{catch}=2.83\). Press Play to watch the no-drag and drag height graphs trace out together.
2. Error table: how wrong is Double It and Square It?

If a tennis ball is thrown straight up and caught at the same height, then the ordinary no-drag rule gives \(H=(2T)^2\). The table compares that estimate with the tennis-ball drag model for catch times from ordinary sculpture tosses up to the practical 7-second ceiling.

Catch time T Double/square height Drag-model max height Error Error
sec ft ft ft inches
1.00 4.00 4.00 -0.00 -0.00
2.00 16.00 16.00 -0.00 -0.03
2.83 32.04 32.01 -0.02 -0.25
3.00 36.00 35.97 -0.03 -0.35
4.00 64.00 63.84 -0.16 -1.96
5.00 100.00 99.38 -0.62 -7.38
6.00 144.00 142.21 -1.79 -21.49
7.00 196.00 191.66 -4.34 -52.08
The rule is within 6 inches until about \(T=4.83\) seconds, which corresponds to a no-drag estimate of about 93 feet. It is within 1 foot until about \(T=5.43\) seconds, about 118 feet, and within 2 feet until about \(T=6.12\) seconds, about 150 feet.
Past that, the exponent really wants to be slightly less than \(2\), but then we lose the beautiful simplicity of Double It and Square It. For all reasonable sculpture-scale tosses, the rule is usually off by inches. Even at the extreme racket-hit ceiling near \(T=7.2\) seconds, the model says the error is only about 5 feet.
3. Hardest-hit ball: why 7 seconds is a practical ceiling

The fastest recorded tennis serve is about 240 ft/sec. If a tennis ball could be hit straight upward at that speed and air resistance did not exist, it would stay in the air for about 15 seconds and rise about 900 feet.

But a tennis ball is fuzzy, light, and draggy. In this quadratic-drag model, the same straight-up hit would stay in the air for only about 7.2 seconds and rise about 200 feet.

That means \(T_{catch}=7\) seconds is a useful practical ceiling for this page. If the world’s fastest serve, aimed straight up, only lasts about 7.2 seconds in the model, then my underhand sculpture tosses are not going to produce anything much longer than that.

Source note: Guinness World Records lists Sam Groth’s serve at 263 km/h, or 163.4 mph. The conversion is about 240 ft/sec.

4. The model: how the equations are solved

Here is the short version of the calculation behind the graph. I am measuring height above my hand, so in both models I take the starting height to be \(h_0=0\).

No air resistance. Start with constant acceleration: \[ \frac{dv}{dt}=-32. \] Integrating gives \[ v(t)=v_0-32t, \qquad h(t)=v_0t-16t^2. \] The maximum height occurs when \(v(t)=0\), so \[ t_{top}=\frac{v_0}{32}. \] If the ball is thrown and caught at the same height, then \(h(T_{catch})=0\), which gives \[ v_0=16T_{catch}. \] Therefore \[ t_{top}=\frac{T_{catch}}{2} \] and the maximum height is \[ 16\left(\frac{T_{catch}}{2}\right)^2=(2T_{catch})^2. \]

For the tennis-ball model, I use quadratic drag. The force from air resistance points opposite the motion, so the velocity equation is

\[ m\frac{dv}{dt}=-mg-\frac12\rho C_D A\,v|v|. \] In feet and seconds, this becomes \[ \frac{dv}{dt}=-g-Kv|v|, \qquad g=32, \qquad K\approx0.00615. \]

On the way up, \(v>0\), so the equation becomes \[ \frac{dv}{dt}=-g-Kv^2. \] This one separates:

\[ \frac{dv}{g+Kv^2}=-dt. \] Integrating gives \[ \arctan\!\left(v\sqrt{\frac{K}{g}}\right)=q-\sqrt{gK}\,t, \] where \[ q=\arctan\!\left(v_0\sqrt{\frac{K}{g}}\right). \] So the upward velocity can be written as \[ v(t)=\sqrt{\frac{g}{K}}\tan(q-wt), \qquad w=\sqrt{gK}. \]

Now the top of the toss is easy to locate: the ball reaches its maximum height when \(v(t)=0\). Since \(\tan(0)=0\), this happens when

\[ q-wt_{top}=0, \qquad\text{so}\qquad t_{top}=\frac{q}{w}. \] Integrating the velocity gives the height on the way up. The cosine appears because the integral of tangent involves \(\ln(\cos)\): \[ \text{maximum height with drag} =\frac1K\ln\!\left(\frac{1}{\cos q}\right). \]

The last step is to make the drag model match the same measured throw-to-catch time. The downward part gives a falling time of \[ \frac{\operatorname{arcosh}(\sec q)}{w}. \] So the total catch time is

\[ T_{catch}=\frac{q+\operatorname{arcosh}(\sec q)}{w}. \] In the graph, I choose the value of \(q\) that makes this equation match the selected \(T_{catch}\). Then I use that same \(q\) in the maximum-height formula above.
  • Mass \(m=57.7\) g and diameter \(d=6.7\) cm, roughly the middle of the regulation tennis-ball range.
  • Air density \(\rho=1.20\text{ kg/m}^3\).
  • Drag coefficient \(C_D=0.55\), a central estimate for a tennis ball.
  • Vertical toss only: no sideways motion, spin, lift, wind, or changing drag coefficient.
  • The ball is thrown and caught at the same height.