“Not to mention the most important reason for climbing a mountain…”
“And that is?”
”Because it’s there.”
James T. Kirk versus El Capitan
For the purpose of examining physics (good or bad) in popular cinema, there could be no better choice than the worst of the ‘Trek films (until Nemesis, at least), Star Trek V: The Final Frontier (aka The One Where Kirk Was Going to Fight Giant Rock Monsters But Didn’t Because They Ran Out of Money). I’ll be examining a scene at the beginning of the film and assessing whether or not the events that occur therein are or are not physically viable using the basic Newtonian physics.
The scene at hand begins with Captain Kirk free-climbing the Yosemite mountain El Capitan (because, of course, with Shatner behind the director’s chair and story development, even the mountain’s name has to remind us of his character). Bones is on the ground in the park watching through a pair of 23rd century binoculars and mumbling angrily to himself (as he often does) as Spock flies over to Kirk via a nifty pair of rocket boots. The crew of the USS Enterprise NC-1701-B is on shore leave, and this is apparently how our three favorite heroes spend their downtime. Kirk is stuck in a spot where he can’t find a higher foothold and, while receiving some unwanted climbing advice from his first officer, slips and begins to plummet towards the ground. Thanks to some horrendous, out-dated-even-in-1989 rear-projection shots, we witness Spock invert himself and use his jet boots to propel towards the falling Kirk. The son of Sarek grabs his Captain’s leg, stopping Kirk with his face mere inches from the ground below.
What does this have to do with physics, you ask? I’ll tell you: a little thing called ‘momentum’ (you may have heard of it) would normally make such a heroic feat an impossibility. Linear momentum (referred to simply as ‘motion’ by Newton and mathematically known as the variable ‘P’) is an easy concept to understand. It’s simply the property of an object in motion calculated by multiplying said object’s mass and its velocity (that is to say that P=mv). That’s all well and good, but the aforementioned scene begins to appear incredulous when we take into account Sir Isaac Newton’s third law of motion: the law of reciprocal actions. That is, for any force there is always an equal and opposite reaction. From this, the law of conservation of momentum is derived, which states that the total momentum in a system is always conserved. Like matter, you cannot simply create or destroy momentum. This means that Spock could not simply catch Captain Kirk inches from the ground and effectively stop his fall. Kirk has momentum (a very large one at that), as does Spock. Both of their momentums are in the same direction, so unless an outside force is acting on Spock to give him a momentum in the upward direction that is equal in magnitude to Kirk’s, they would both face a grim, splattered demise.
So let’s delve into the math behind this bad boy, shall we?
What this boils down to is a simple total momentum equation for this closed system (which provides a truly perfect example of an inelastic collision because the two objects will, for all intensive purposes, become “one” afterwards [and I don’t mean Spock will bestow his katra upon Kirk]): Kirk’s momentum and Spock’s momentum should, when added, equal zero if we are to truly believe that Spock could have saved Jim.
ΣP = PKirk + PSpock
which, when transformed slightly becomes
0 = mKirkvKirk + mSpockvSpock
To solve this, we’ll need the masses of both William Alan Shatner and Leonard Simon Nimoy and both characters’ velocities at the moment Spock grabs Kirk’s ankle. Shatner measures in at 5’9.5” or 1.77m and Nimoy at an impressive 6’1” or 1.85m. A quick glance at a Body Mass Index chart estimates Nimoy’s mass at approximately 70.5kg (assuming he is right in the middle of the “healthy weight” category) and Shatner’s at 77.3kg (assuming he is on the better side of the “overweight” category because this is, after all, 1989, though he probably weighs about as much as he did on the Original Series due to the increase in his gut’s mass being countered by the decrease in his hair’s mass). As for the velocities, Kirk’s is 186.2m/s because he is partaking in a freefall and therefore is experiencing the average acceleration of gravity (9.8 m/s2, which I then multiplied by 19s because that is how long Kirk has been accelerating when Spock catches up to him). Spock, on the other hand, has to accelerate to catch up to Kirk as he doesn’t begin his descent until 4 seconds after Kirk falls and it takes him 15 seconds to reach the good Captain’s ankle. Based on this, Kirk is already 39.2m below Spock’s position when Spock begins his descent. Based on the 19 seconds it took Kirk to reach the bottom (or one inch from the bottom, rather) at his velocity, Spock is 186.2m up El Capitan (a mere fraction of the mountain’s 910m height, which is misleading because it certainly appeared that Kirk was at least halfway to the peak when he fell) when he starts propelling downwards. Since it would be impossible to determine the rate at which Spock would be accelerating to reach Kirk in time, I’ll just calculate his average velocity (hey, this is starting to sound like one of those “a man is being chased by a bear” problems we used to do).
Since v=d/t, Spock’s average velocity ends up being 186.2m/15s or 12.41m/s.
Now we can take all of this and plug it back into our handy equation we figured out earlier:
0 = mKirkvKirk + mSpockvSpock
0 = 77.3kg*186.2m/s + 70.5kg*12.41m/s
which, when solved looks like this:
0 = 14393.26kgm/s + 874.9kgm/s
or
0 = 15268.16kgm/s
which, as we should realize, is not a true statement.
This means that the total momentum in this system consisting of Kirk and Spock would actually be equal to 15268.16kgm/s the moment before they hit the ground, which would certainly not stop either of them but, instead, result in a horrific splattering that even the on-looking Bone’s couldn’t cure (after all, he’s a doctor, not a sanguine-fluid-collecting-bucket).
From all of this, it is safe to conclude that Spock could not have stopped Kirk’s freefall by simply catching hold of his leg. However, this position neglects the fact that Spock is wearing boots with thrusters attached to them (futuristic thrusters, to boot [pun intended]). Perhaps he caught Kirk and then activated the thrusters in the upward direction, giving himself a velocity that, when multiplied (quicker than a tribble, even) by his mass, would result in a momentum for Spock that is equal and opposite to Kirk’s own momentum (although this is absolutely not the case because we see that Spock is still upside down until long after Kirk stops falling, and his boots could not propel him upward from this position because the thrusters are shown to only shoot outwards from the bottoms of the boots, but let’s for a moment ignore this fact). So what kind of acceleration are we talking here? Exactly at what rate must the boots accelerate for Spock to safely bring Kirk to a stop? Well, it’s a simple question really. If the boots are giving Spock and equal and opposite momentum to Kirk, than our equation from earlier will now be true:
0 = mKirkvKirk + mSpockvSpock
and since we know that Kirk is still freefalling,
0 = 14393.26kgm/s + -14393.26kgm/s (she’s negative because we made the distinction earlier that down is the positive direction)
Some quick algebra tells us that Spock’s new velocity must be –204.16m/s, if he is to save Kirk. This means that the force of those thrusters will have to be so great as to instantaneously change Spock’s velocity from the 12.41m/s necessary for him to catch up to Kirk to the –204.6m/s necessary for him to cancel out Kirk’s momentum (an amazing change of 217m/s)! Now we simply bust out one of the legendary, handy-dandy big four equations and solve. The only problem is that each of the big four requires either the elapsed time (t) or the distance traveled (y, since we’re dealing with a strictly vertical space). So, even though the stop appears instantaneous, let’s give Spock the benefit of the doubt and say that this thruster propulsion occurred in .01s
vf = vi + at
and since we’ve got everything we need save acceleration:
-204.16m/s = 12.41m/s + a*.01s
This results in an a value of –21657m/s2. I don’t know about you, but I think that’s awesome. While there’s no way any propulsion system could ever function so ridiculously powerfully (even nuclear wessels couldn’t achieve such a speed), maybe we won’t discover it until the 23rd century (unless Mr. Scott travels back in time and tells us about it sooner).
Now, this isn’t all fun and games here. These calculations aren’t without error. For one, air resistance isn’t taken into effect. This would surely slow both Kirk and Spock’s respective velocities and therefore lessen the magnitudes of their momentums. Furthermore, by using the average gravity of Earth, I’m miscalculating Kirk’s true velocity. In reality, he wouldn’t continue to accelerate indefinitely, but instead reach his terminal velocity. This of course throws off his momentum’s magnitude. The same goes for my foregoing of determining Spock’s acceleration as he catches up with Kirk (which cannot be determined with the information available) and using his average velocity in its stead. Spock would likely be traveling even faster when he reaches Kirk (although that just means that the impulse he feels when he hits the ground will be greater, as he will still not be able to stop the fall). But despite these errors, the conclusion remains the same.
To wrap this up, this scene is not physically viable in the least. If the scene were to end in the way it would in real life, Kirk and Spock would surely end up in Pike Boxes. As filmed, Spock simply grabs his Captain’s leg (albeit off-screen) and the two suddenly stop falling (with Kirk dangling mere inches from the ground). Physics says this can’t happen, because each of their momentums would simply be added, resulting in the two of them continuing to fall (until they hit the ground, which would become part of the system, absorbing much of their momentum and turning this collision from elastic to inelastic). Unless another force acts on Spock to propel him upwards the moment he grab’s Jim’s leg (like his jet boots), there’s no way for this to happen. And even so, the tendon’s in the human leg would certainly rip, resulting in Kirk still liquefying on impact while Spock rocket’s upwards with his late Captain’s leg in hand: “Your leg is, and always shall be, my friend.”
Captain, I do not think you realize the gravity of your situation.
”On the contrary, gravity is foremost on my mind.”
References
"BMI Chart." Boston Medical Center. 19 May 2009.
Star Trek V: The Final Frontier. Dir. William A. Shatner. Perf. William Shatner, Leonard Nimoy, and DeForest Kelley. DVD. Paramount Pictures, 1989.
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