The question
A Sportsnet reporter, Shawn McKenzie, recently asked a question that’s surprisingly tricky to answer:
could someone standing on the CN Tower throw a baseball all the way to third base at the SkyDome?
From the CN Tower’s observation deck or EdgeWalk the baseball field looks close. Close enough to be a tempting target for a baseball throw…
Sportsnet turned the question into a great 5-minute video you can watch on YouTube:
(It’s also on Instagram here: https://www.instagram.com/p/DPRkyJ_EaRE/?igsh=Z2JuMXB5YzJhcWc1)
They included a short clip from what was a 30+ minute interview. We had a great conversation. For anyone interested, I thought I’d post more details about the physics of this scenario.
(Personal aside: I worked as a host at the CN Tower during my undergrad degree. Hosts are the people who help visitors everywhere around the site – including the elevators rides. I enjoyed it. I recall looking down at the baseball field when the dome was open. However, it was also part of my job to ensure people didn’t throw objects from the lower observation deck, which used to be open to the air with a metal mesh…)
Set up: the key information we need
To do this calculation, a few key facts need to be clarified:
- Characteristics of a baseball,
- Height, i.e., where the person is positioned at the CN Tower,
- Target distance, i.e., from the CN Tower to SkyDome’s third base, and
- Speed of the throw, e.g., by a regular person and by a professional.
With this information we can estimate an answer.
A baseball’s physical characteristics are described by Major League Baseball official documentation here (section 3.01):
- Mass: 5 to 5.25 ounces, which I converted to 145 grams
- Circumference: 9 to 9.25 inches, which I converted to a radius of 3.7 cm
- Shape: sphere, but not perfectly smooth.
I used 365 m for the height of the person throwing the baseball. This is the height of the EdgeWalk, located on top of the observation section of the CN Tower. This seemed to be the most practical premise since it is where visitors can walk around outside. (I did the EdgeWalk years ago with a friend and I would do it again!)
263 meters was used as the distance of the target of third base. This is difficult to determine precisely and introduces a key uncertainty in answering the question. My first attempt at a rough estimate was to examine Google Maps and Google Earth. This suggested a distance between 200 m and 270 m. The dome is not open in most of the images but it’s worth noting that third base is the most distant of the bases from the CN Tower. Satellite images can be misleading due to viewing angles and distortions. A direct measurement is preferable.
The segment producer measured the angle at the ground from third base (our target) to the EdgeWalk, which has a known height. From this, some trigonometry enables us to calculate the distance to be 263 m. This seems plausible and within the range suggested from the Google images. There are areas within the baseball diamond and stadium seats that would be closer to the CN Tower and easier to reach. Any distance greater than 200 m would likely be inside the SkyDome field and stands. As it turns out, that’s a much more achievable target.
Finally, we need to know how fast a person can throw a baseball.
- For a ‘regular person’, I used 90 km/h (25 m/s or 56 mph). This might be generous, but let’s think positively.
- For a professional, I used 162 km/h (45 m/s), which is 100 mph. This seemed reasonable since the best pitch speed recorded in the current MLB is 170 km/h. This is roughly double the speed of a regular person.
Also: throw angle
The throw angle is also very important. For each throw speed, I used the angle that maximizes the horizontal distance. When throwing from a large height, the angle needs to be shallow (small) to maximize the horizontal distance.
Physics of the throw
The most important factors affecting the throw distance are gravity and air resistance.
Gravity is constant at the CN Tower: objects are accelerated downward at 9.81 m/s2.
The density of air varies by height and in time. I used a typical value of 1.225 kg/m3.
To illustrate the importance of these two factors in combination, imagine simply dropping the baseball from the EdgeWalk height of 365 m.
Without considering air resistance, the ball would take about 8.5 seconds to reach the ground. However, once we add air resistance, the ball would take about 12 seconds.
For comparison, dropping the ball from the same height on Mars, which has one third the gravity of Earth, would take 14 seconds to reach the ground without air resistance and 20 seconds with (Earth’s) air resistance.
Results: ignoring air resistance
A first year undergraduate physics student should be able to analyze the motion of the baseball and calculate the range without air resistance. This is not realistic. However, let’s start there.
This result shows that a professional can throw a baseball to third base. Perhaps even reaching the opposite side of the SkyDome stadium! A regular person could throw the baseball onto the field, but falls short of third base.
Now let’s consider the effect of air resistance. It may be more dramatic than some expect.
Results: with air resistance
There are a few challenges to doing this calculation with air resistance.
The drag force created by the air is (very) speed dependent. As a result, you can’t do one calculation. Instead, I had (python language) code perform a calculation every 0.01 seconds of flight time to account for the changing speed and drag.
There is another decision: what model of drag should be used? I used the equation found in first year undergraduate physics textbooks:
For this situation, it is appropriate.
In the equation:
- A is the area of the object in the direction of motion, a circle with a 3.7 cm radius
- ρ is the density of the fluid (I’ve taken that to be 1.225 kg/m3)
- v is the velocity of the object (baseball)
- C is the drag coefficient. I used 0.38.
A typical C value for smooth spheres is 0.5. However, a baseball’s surface is rough and (significantly) it has raised stitching. The physics literature has measured a range of results for this value, e.g., Kensrud & Smith (2010), especially Figure 7, and Kagan & Nathan (2014). I used 0.38 as a conservative estimate.
Here are the results using those values:
Neither the regular person or the professional was able to reach the target. The professional might have been able to throw the baseball into the SkyDome, e.g., somewhere in the stands nearest to the CN Tower. That’s still impressive!
A logical question to consider is:
What throw speed IS necessary to reach third base, at a distance of 263 m?
Answer:
- Without air resistance: 104 km/h (65 mph)
- With air resistance: 300 km/h (186 mph)
It’s unrealistic for someone to throw a baseball at 300 km/h. These results suggest that someone might be able to throw a baseball from the CN Tower EdgeWalk inside the SkyDome. However, it’s very unlikely they could reach third base.
Other solutions?
If 300 km/h seems impossible for a person, what else could we do?
Get higher
Additional height doesn’t help the horizontal range much. Even if you somehow stood on the very tip of the CN Tower spire (not recommended, especially in stormy weather), a height of 553 m doesn’t extend your throw by more than several meters. Air resistance and gravity are too much of a constraint.
Spin
There is some range to be gained by ensuring the throw includes backspin. The magnus force created by spin can create a form of lift. Experimental analysis of backspin on baseball flights show that it is a difficult topic. Spin can help extend horizontal range (e.g., Nathan, 2008; Alaways & Hubbard, 2010; Kensrud & Smith, 2010). However, this also increases drag. The scale of the benefit would not be enough to make up the distance shortfall we saw in the earlier calculations.
Moving to a different location
Since drag from air resistance is proportional to air density, we could move to a higher altitude, where the air density is lower. Moving the CN Tower is not realistic, but we could think about it for fun.
The major city at the highest elevation I found was Lhasa, Tibet. It is at an altitude of 3.65 km and the air density is 0.83 kg/m3. However, it seems this change isn’t enough to meet our goal.
What about the highest elevation in Canada?
Mount Logan in Yukon Territory has a height of 6 km. At this altitude, half of the atmosphere’s mass is beneath you! The air density is 0.6 kg/m3.
Success!
The professional can throw the baseball and hit the target.
All we have to do is move the CN Tower and SkyDome to the peak of Mt. Logan.
Change gravity
What if we could change gravity? e.g., to match the strength of gravity at the surface of the moon or on Mars?
If we magically changed to Martian gravity (3.7 m/s2), a regular person still falls short (173 m); however, the professional gets close enough that I think we can claim potential success: 247 m.
If we push further and consider Lunar gravity (1.625 m/s2), a regular person still falls short but likely reaches the inside of the stadium. Maybe the baseball even gets onto the field.
Actual solution: wind
In the real world, a lucky gust of wind is the only way I think a professional pitcher can throw a baseball from the CN Tower EdgeWalk and it reaches third base in the SkyDome. The effect of wind is not trivial to calculate, and it varies in time and altitude. However a strong and sustained gust of wind could give the extra distance needed.
But this situation raises a philosophical question:
if the wind does much of the work, can we really say a person threw the ball to third base?
How far do you think you could throw a baseball from the CN Tower?
Further reading
There have been quite a few people interested in the physics of sports. Published papers about baseballs go back at least several decades. Here are a few highlights for anyone interested in reading physics literature about the details.
D. Kagan and A. M. Nathan, “Simplified models for the drag coefficient of a pitched baseball,” Phys. Teach. 52, 278–280 (2014). https://doi.org/10.1119/1.4872406
A. M. Nathan, “The effect of spin on the flight of a baseball,” Am. J. Phys. 76, 119–124 (2008). https://doi.org/10.1119/1.2805242
A. M. Nathan, “The physics of baseball: What’s the deal with drag?” Phys. Teach. 53, 332–335 (2015). https://doi.org/10.1119/1.4928349
L. W. Alaways and M. Hubbard, “Experimental determination of baseball spin and lift,” J. Sports Sci. 19, 349–358 (2001). https://doi.org/10.1080/02640410152006126
J. R. Kensrud and L. V. Smith, “In situ drag measurements of sports balls,” Procedia Eng. 2, 2437–2442 (2010). https://doi.org/10.1016/j.proeng.2010.04.012
J. R. Kensrud and L. V. Smith, “Drag and lift measurements of solid sports balls in still air,” Proc. Inst. Mech. Eng. Part P: J. Sports Eng. Technol. 232, 255–263 (2017). https://doi.org/10.1177/1754337117740749