6. The leaning tower of Pisa
There is an old story that Galileo Galilei once climbed to the top of the Leaning Tower of Pisa to simultaneously drop two cannon balls to the ground below. According to this tale, Galileo wanted to show that even though one of the cannon balls was heavier than the other, both cannon balls would hit the ground at the same time. This result would demonstrate his theory that the acceleration of each cannon ball was independent of its mass. While Galileo did describe this experiment in his notes, he never claimed to have performed it and many science historians therefore believe that it never happened. Nevertheless, there is no dispute that, among his other great achievements, Galileo performed the first systematic study of uniformly accelerated motion. As a result of his careful investigation, Galileo determined that objects near the earth will fall with constant acceleration, and this constant acceleration is independent of the object’s mass. Galileo was born in 1564, and now almost 450 years later, his research directly impacts how our characters fall in Azeroth and the Outland!! To see the connection between Galileo’s research and the World of Warcraft, we need to examine the work of another great scientist, Isaac Newton. Newton was born in Jan of 1643, almost exactly one year after Galileo died. Like Galileo, Newton made an enormous contribution to modern science including what is now known as ‘Newton’s Second Law’. This law states that a force on an object is equal to the mass of the object multiplied by its acceleration, F = m a. Combining Newton’s Second Law with Galileo’s observation that objects fall with a constant acceleration independent of mass, we obtain,
Here, ‘g’ is the constant acceleration due to gravity, and the acceleration ‘a’ is the second derivative of the position ‘z’ with respect to time. The negative sign in front of the mass m indicates that the force is down in the negative z direction. Dividing both sides of equation 1 by the mass yields,
Integrating equation 2, we obtain the velocity as a function of time,
where, by our choice, vo = 0 is the initial velocity. Integrating equation 3, we obtain the position as a function of time,
where zo = htot is the initial position. Using these equations we see that an object falling with constant acceleration will have a velocity that increases linearly with time, and a position that is a quadratic function of time. Now, as anyone who has put their hand out the window of a speeding car knows, in addition to the force of gravity, a falling object will experience a force associated with wind resistance. The force of wind resistance on an object is opposite to the direction of motion and increases as the velocity increases. So, as an object falls, the force of gravity will accelerate the object towards the ground. As the velocity of the object increases, the force of wind resistance will increase in opposition to the force of gravity. Eventually, the object will be falling fast enough so that the force of wind resistance will exactly balance the force of gravity. At this point the object has reached its terminal velocity. As a point of reference, a skydiver falling in a spread eagle position will have a terminal velocity of about 120 Mph. Now, once an object reaches its terminal velocity, it will no longer be accelerating, a = 0, and equations 2-4 must be modified. Starting with the expression for acceleration,
Integrating equation 5 will now yield a constant velocity equal to the terminal velocity,
After integrating equation 6 we obtain the position as a linear function of time,
There are a number of possible ways to program these equations into the World of Warcraft. The model I'll investigate splits the process of free falling into two phases. During the first phase, a character will fall with constant acceleration as described by equations 2 - 4 where the force of wind resistance is ignored as a simplification. So, during this first phase, a character will accelerate from some initial velocity to some terminal velocity. Phase two of a free fall descent begins once a character has reached the terminal velocity. For this phase of the descent, equations 5-7 will govern the dynamics. You can check for your self that this two phase model is at least approximately true if you fly high above a body of water in the Outland and then dismount. At first you will fall slowly, picking up speed, until you reach what appears to be a maximum rate of descent. The trailing plume from Prym’s Mana-Sphere shoulders provides a serendipitous verification of this. The spacing between periodic features of the plume structure is proportional to Prym’s velocity. This spacing increases as Prym free falls eventually reaching a maximum value corresponding to the terminal velocity.
To get a better feel for this two phase free fall model it helps to look at the figure below,
In the image above, we see Prym just beginning a free fall starting high above a small lake in Nagrand. This lake in Nagrand is shown in the image below,
I like to fall into the water because it saves on my repair bill. During the first phase of the descent Prym undergoes constant acceleration for a distance hacc in a time tacc. At this point, Prym reaches his terminal velocity and falls the remaining distance hc to the water in a time tc. If we solve equation 4 above for hacc we get,
And solving equation 7 for hc we get,
Now, the total height htot = hacc + hc, and substituting with equations 8 and 9 yields,
where we have used the total time ttot = tacc + tc.
In equation 10, g, tacc and vterm are constants that are built into WoW. Equation 10 is also an expression for a straight line, where r = (g/2 tacc^2 – vterm tacc) = hacc is a constant intercept, and s = vterm is a constant slope. So all we have to do to verify our two phase model is to measure a series of free fall heights, htot, along with the corresponding free fall times, ttot, and then plot the result. If we get a straight line, we know that we either have the right model for free fall dynamics or a very good approximation.
So, to figure out how high Prym is before I measure the time it takes to free fall into the water below, I carried out a series of timed vertical flights. For these flights, I started at the landing site and flew straight up in the vertical direction on my epic bird with no crop. Eventually, I hit the invisible ceiling that sits above all of the Outland. I knew I hit the ceiling when my bird squawked. I repeated this vertical flight six times and the average time it took to fly straight up to the invisible ceiling was 23.07 seconds. Assuming that the vertical flight speed is the same as the horizontal flight speed of 53.2 Mph (as measured in part four of ‘Fun with the Minimap’) we can estimate that the invisible ceiling in Nagrand is approximately 1800.1 feet above my landing site.
I then flew straight down from the ceiling to the landing site. The average time over six flights was 23.05 seconds. Here we see that birds fly up as fast as they fly down. If I use a vertical flight speed of 53.2 Mph, I find that the ceiling is 1798.8 feet above my landing site. Finally, I used the space bar to levitate vertically, and the average time over six levitations was 23.11 seconds. I attribute the slight difference between the levitation time and the vertical flight time to the timing of the birds squawk when it collides with the invisible ceiling with a horizontal versus a vertical orientation.
To verify that the vertical flight speed is indeed the same as the horizontal flight speed of 53.2 Mph, I performed one last measurement of how high the invisible ceiling was above the landing site. For this last measurement, I flew roughly a third of the way up to the ceiling and landed on a floating island. I got off my bird, cast slow fall, and then remounted. I was then able to fly up to the ceiling while slow fall was still in effect and dismount. I then slow fell from the ceiling all the way to my landing site with an average time of 87.5 seconds. Using the slow fall descent rate of 14.04 Mph measured in part five of ‘Fun with the minimap’ I obtain 1802.3 feet for the height of the ceiling above the landing site. This confirms that the invisible ceiling is indeed about 1800 feet above the landing site and the vertical flight speed (up, down and ‘levitate’) of an epic bird is the same as the horizontal flight speed. So, all I need to do to obtain a measure of htot and ttot required to test our free fall model is to first fly vertically upward for a predetermined amount of time. My height after this predetermined flight time is given by the product of the vertical flight time with the vertical flight speed. Then, I simply dismount in midair and measure the time it takes to free fall into the water at my landing site. I performed htot and ttot measurements by flying vertically up for times that ranged from 5 seconds up to 22 seconds in increments of one second. I repeated each measurement twice for each vertical flight time. For example, if I fly vertically up for 20.08 seconds, Prym is 1566.8 feet high. A vertical free fall into the water from this height takes 11.01 seconds. The results of my measurements are shown in the figure below and are fit with a straight line,
Here we see that the data does indeed lie along a straight line and confirms that we either have the correct model for how free falling is simulated in the World of Warcraft, or a very close approximation. The slope of this line is 53.189 meters per second. Converting the slope to Mph, we see that the terminal velocity is 119.0 Mph. Using,
and averaging over all the ttot and htot pairs, we find that the acceleration due to gravity is about 12.7 meters/ sec^2. For comparison, the acceleration due to gravity on the earth is 9.8 meters/sec^2. We can also use,
to find that the distance it takes to reach terminal velocity is hacc = 365.5 feet (111.4 meters) and time it takes to reach terminal velocity is tacc = 4.2 seconds. These results are summarized in the table below,
|Gravitational Acceleration (g)||12.7 m/s^2|
|Terminal Velocity (vterm)||119.0 mph|
|Distance to reach vterm (hacc)||365.5 feet|
|Time to reach vterm (tacc)||4.2 sec|
So next time you find yourself plummeting towards the ground, you can take solace, meager as it might be, in the knowledge that you are not unlike Galileo’s purported cannon balls, crashing down from the Leaning Tower of Pisa.