Speed of Falling Saruman: Lord of the Rings
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Abstract
The purpose of this analysis is to determine the speed of Saruman after falling from the top of the tower Orthanc.
Introduction
This scene is from Chapter 4, "The Voice of Saruman," in the extended edition DVD of The Return of the King (2003). In this scene, Saruman the white wizard is trapped in his tower and is talking to Theoden the King of Rohan (as well as other members of the Fellowship of the Ring). He is then stabbed several times by Wormtongue, his former ally. After the stabbing, Wormtongue is shot with an arrow by Legolas, the elf in the Fellowship. Wormtongue falls backwards but Saruman topples off the edge of Orthanc onto a spiked wheel.
We are going to determine two values of the speed of Saruman using the formulas
v2 = vo2 + 2as
and
v = vo + at
Where
- v is the speed as he hits the spike (in m/s)
- vo is the initial speed, which is equal to 0 m/s
- a is the acceleration, which we are assuming is g (9.8 m/s2). We are also neglecting air resistance.
- s is the distance Saruman falls from where he is standing to the wheel
Procedure
- Find the scene in the movie entitled "The Voice of Saruman" (Chapter 4). Locate the part where Wormtongue stabs Saruman.
- Using a stopwatch, measure the time from the moment he falls until the moment he hits the spike. Do 20 trials, then find the average of the trials. (See data table in Analysis.)
- Locate the part of Scene 4 where Saruman is standing on top of the tower and you can see from the level that he is standing to the top.
- Using a computerized ruler, measure the height of Saruman standing on top of the tower in centimeters. Then, measure the height from the level where he is standing to the top of the tower.
- According to the International Movie Database, the height of Saruman (played by Christopher Lee) is 6'5" or 1.96 meters. According to Wikipedia, the height of Orthanc is 500 feet or 152.4 meters. Using a proportion, determine the distance from the level where Saruman is standing to the top of the tower, which is equal to 15.33 meters.
- Locate the part of the scene where you can see the wheel that he falls onto and Gandalf (on his horse) next to it.
- Since the height of the wheel is twice the height of the horse and rider, we can double the height of Gandalf on his horse. According to Wikipedia, his horse, Shadowfax, is an Andalusian, with a height of 15.2 hands or 1.54 meters. According to the International Movie Database, the height of Gandalf (played by Sir Ian McKellen) is 5'11" or 1.8 meters. Adding the height of the horse to half his height gives us a height of 2.44 meters. Then we multiply by two to obtain the height of the wheel which is 4.88 meters.
- To determine the distance that Saruman falls, add the distance from Saruman to the top and the height of the wheel and then subtract that from the total height of the tower. Thus 152.4 m − (15.33 m + 4.88 m) = 132.19 meters that Saruman falls.
| Trial | Person 1 Time (s) |
Person 2 Time (s) |
|---|---|---|
| 1 | 7.46 | 7.34 |
| 2 | 7.05 | 7.03 |
| 3 | 7.34 | 7.16 |
| 4 | 6.89 | 6.85 |
| 5 | 6.84 | 6.99 |
| 6 | 7.21 | 6.84 |
| 7 | 7.18 | 7.16 |
| 8 | 7.08 | 6.91 |
| 9 | 7.21 | 7.23 |
| 10 | 6.99 | 7.10 |
Analysis
The first method for finding the speed of Saruman as he hits the wheel is to use the equation
v = vo + at
where vo is 0 m/s and a = g = 9.8 m/s2.
The average of the trials is 7.09 seconds. Substituting the known values into the equation,
v = 0 m/s + (9.8 m/s2*7.09 s) = 69.5 m/s (155.5 mph).
The second method for finding the speed of Saruman is
v2=vo2+2as
where vo is 0 m/s, a = g = 9.8 m/s2, and s = 132.19 m. Substituting the known values into the equation, v2 = 2*9.8 m/s2*132.19 m = 2590.92 m2/s2. Thus v = 50.9 m/s (113.9 mph).
Conclusion
Based on this analysis and by using our first method, Saruman's velocity is 69.5 m/s (155.5 mph) as he hits the spike. However, his true velocity is somewhat less than this because air resistance decreases his acceleration. By using our second method, Saruman's velocity is 50.9 m/s (113.9 mph). Of the two methods, the second is the better approximation because it uses the height that he falls even though the acceleration is still slightly less than g.
Sources of Error
One source of error for this analysis is that we neglected air resistance. Air resistance would decrease Saruman's acceleration which would lead to a smaller final velocity.
Another source of error is that while Saruman is falling, he is also rotating. Since he is rotating, the acceleration is not constant due to change in drag because of change in his projected area.
Gina Castellano, Jenna Conversano -- 2005
Physics on Film