Mission:Impossible IV – Tom Cruise’s 100-meter, 100-mph Drop (RPMA)

See the universe through a brand new set of
eyes. In the final minutes of Mission Impossible:
Ghost Protocol, Tom Cruise (as agent Ethan Hunt) must send a signal that will disarm
a nuclear missile headed for San Francisco. Cruise is located on the upper floor of a
very tall parking garage in Mumbai. The computer briefcase needed to disarm the missile has
fallen to the concrete floor of the garage, 100 METERS BELOW! (My laptop is a goner when
it falls just one meter … onto carpet!) We know that Cruise will save the day if he
can just get down to that briefcase in the few seconds before the missile — a Tom Cruise
Missile? — reaches San Francisco! Let’s see what happens.. Yikes! DOES Cruise save the day? Can he crawl
out of the car, get to the briefcase, and disarm the nuclear missile? Well, if Cruise
can survive THAT crash — airbag or not — then I figure he can do just about anything! The key issue is the car’s impact speed on
the concrete floor. We’re going to need physics and mathematics to find that speed. So, let’s
symbolize speed with the letter “v,” in accord with global tradition. We choose “v” because
of speed’s association with velocity, v-vector. Physics offers several ways to find speed.
Which way we choose depends on what information we have, i.e., what data are “given.” The
crucial piece of information given in this problem is the initial height of the car above
the garage floor. Let’s call this height “h.” We know that h=100 meters because we are
told so later in the movie. Take a look… Is a 100-m drop also consistent with what
we actually SEE happening to the car as it falls? Well, if we time how long the car takes
to fall, we can use a very basic physics formula to calculate the fall distance. The formula
is distance=1/2(acceleration due to gravity) x (fall time)^2. d=1/2gt^2. I made five distinct measurements of the fall
time and got an average value of 5.3 seconds. The distance formula then gives d=140 meters:
d=0.5(9.8m/s^2)(5.3s)^2=140 meters But we’re TOLD that the fall height is 100
meters, not 140, so let’s stick with 100. Besides, the tallest such parking structure
on Earth has TWENTY stories. (It’s not in Mumbai, it’s in Wolfsburg, Germany.) 20 stories
times 3 meters per story (that’s about 10 ft per story) is only 60 meters. 100 meters
sounds barely plausible. But that IS what we are told! 140 meters — what we calculate
— just does not seem possible. So we are “given” that the car falls from
a height of 100 meters. Here comes the hardest part of solving most physics problems: What
law of physics should we apply? What formula involves both fall height (which we’re given)
and fall speed (which we want to find)? Even if we know all the laws of physics, how do
we choose the RIGHT law? The answer is: By EXPERIENCE. Experience solving problems reveals
this: When you want to find speed, first try Energy Conservation. The law of energy conservation says that “initial
energy equals final energy.” Here, this means that energy at the top=energy at the bottom. Only two TYPES of energy are involved as the
car falls: gravitational potential energy and kinetic energy. Here’s the straightforward
— and crucial — translation of these ideas into mathematics: The total energy E at the top=the total
energy E at the bottom PotentialEnergy(top) + KineticEnergy(top)
=PotentialEnergy(bottom) + KineticEnergy(bottom) Potential energy is energy of POSITION; in
this case, that’s the energy of HEIGHT. At the bottom, right before the car hits, the
height is almost zero, so let’s cross out that term. Kinetic energy is the energy of
MOTION. In this case, there’s virtually no motion at the top, just as the car starts
to fall, so let’s cross out THAT term, as well. We cross the zero terms out — rather
than erase them — so that we SEE how energy conservation works, and so that we can show
others that we haven’t neglect any relevant energy. It’s just that two of those energies
happen to be ALMOST ZERO in this problem. We want the speed, v, of the car at the bottom,
just before it hits. That “v” is hidden in the kinetic energy: mgh=1/2mv^2
2gh=v^2 v=the square root of 2gh Let’s not be too eager to plug numbers into
this formula. Notice that there’s no “m” anywhere in the equation. The mass “cancelled out”
in the first step, here. That’s very significant. Energy conservation tells us that the mass
of a falling object (like the car) does not influence its speed. WEIRD! If Tom Cruise
were to bail out of the car, he would fall at the same rate as the car. Light objects,
in free fall, go just as fast as heavy objects. If Galileo could see this, he would weep with
joy. 500 years ago, Galileo REALLY struggled to figure out that the mass of a freely falling
object doesn’t matter, and he worked even harder trying to convince others. Today, we
see it in one line of algebra! By the way, we’re seeing, here, one of the
beautiful features of mathematical physics: “pure” math (the canceling of the m’s) is
telling us something about the “real” world (mass doesn’t matter). This kind of insight
from mathematics happens all the time in physics: Since the variables in our equations describe
physically meaningful entities (like mass and speed), what HAPPENS in the equations
often has physical, real-world, implications. But we can’t see those implications if we
rush to plug numbers into our equations too soon. Here, a rush to calculate would have
kept us from seeing that the m’s cancel. Instead, we held out; we played with the symbols; we
did algebra, not arithmetic. And we were rewarded with an unexpected insight that is much more
generally valuable than the answer to this particular problem. This is a good problem-solving
rule to remember: Do as much symbolic math as you can before plugging-in numbers. All right. Now that we’ve done all the symbolic
algebra we can, let’s DO plug-in some numbers — AND units — into our symbolic formula
for the speed of the fallen car. v=the square root of 2(9.8m/s^2)(100m)
That’s about 45 m/s — v approximately equals 45 m/s. In miles per hour that’s 100 mph!
Tom Cruise (and his car) are moving at 100 mph when they smack into the concrete floor!!! What’s it like to have a 100 mph head-on collision
with something that doesn’t “give”? To find out, click on the Mythbusters link presented
here. But do come back. Contrast these two images… Here is Cruise’s
car from the movie. And here is the car that REALLY had a 100 mph head-on collision! In
the REAL collision, the airbag is irrelevant because the entire front half of the car has
been crushed. Even Tom Cruise is not gonna be able to crawl out of that car and save
San Francisco. Mission impossible? Yes: IMPOSSIBLE MISSION for sure!

3 thoughts on “Mission:Impossible IV – Tom Cruise’s 100-meter, 100-mph Drop (RPMA)

  1. There is an error in all this, the 100m drop comment is originally a 100 feet comment, probably changed depending on what region the release is from. The tower it's based on (accurately) is actually only 48m high, making the 100 feet comment more accurate.

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