Yet, after the credits role, and the movie goers make their way
home, the images that stick with them are not those of kissing,
lunacy, or explosions, but rather the exciting set-pieces involving
mechanics that are continuously on display. From the elevator on which
the film opens, to the bus, where the majority of it takes place, to the subway
on which it concludes, it feels like a 116-minute mechanics course,
albeit an entertaining one. I don't know this for a fact, but I would
suspect that director Jan de Bont took a physics class as a kid and enjoyed it
immensely.
I suppose it is not surprising that the movie features mechanics,
as its title is a key term of kinematics - speed is defined as
the magnitude of velocity. And, when an ex-cop turned
psycho attaches a bomb to a city bus, he programs it with this kinematic
parameter in mind: the bomb is armed once the bus surpasses a speed of 50 mph,
and is set to blow should it ever fall below this value again.
If there were more class time in the Mechanics course that I teach,
I would actually show Speed in class. And, after each
action sequence, I would pause the film to discuss the key concepts of
mechanics on display, and even solve explicitly for some of the unknown
parameters. As this exercise is quite time-consuming, I simply encourage
my students to try this activity on their own.
In the first scene alone, many aspects of mechanics are highlighted when
an elevator filled with innocent people threatens to plummet to the
ground. The periods of free fall experienced by both the
elevator and those inside begs several questions, like "Should
the passengers float upwards?" and "Would they increase their
likelihood of survival if they jumped just before the cabin hits the
ground?" I'll leave readers to consider these on their own.
When the cabin and its contents are supported by a single rope, how much
tension manifests inside it? Does the rope extend, and if so, by how
much? Why does the supporting crane above break? I'll answer this
last one: the tension in the cord creates a large moment (or torque) about the
support structure. The bending moment leads to a local stress that is
larger than the ultimate stress value of the material making up the structure.
We could go on and dissect the mechanics of the entire film in this
fashion, but instead, I would like to focus on two particular action sequences
in some detail. These two sequences occur on the fast-moving bus, and I
always discuss them with my mechanics classes.
Midway through the movie, Bullock's character, the stand-in bus driver,
must attempt a sharp turn at a high speed. When a moving vehicle follows
a circular path, it has an inherent centripetal acceleration that points
towards the center of the path. The magnitude of this acceleration
is v2/R, where v is the speed of the
vehicle and R is the radius of the circular path.
Typically, when a vehicle rounds a bend, it reduces its speed to reduce
the magnitude of this acceleration. A car does this to avoid skidding
outwards during the turn. This would occur if the inertial term of the
car (its mass multiplied by its centripetal acceleration) were to exceed the
maximum static friction between the tires and the road. To help cars
avoid skidding, curved roads are often inclined upwards at some angle - the
result of this feature is that the inertial term must overcome both the
static friction as well as a small component of the car's weight in order for
it to skid.
As our friends in Speed prepare
for a sharp turn, their concern is not that they will skid, but that they will
roll over. A typical sedan has a center of mass that is quite close to
the ground, such that it will always skid before it rolls over. A bus,
however, has a center of mass that is fairly elevated, and is at risk of
rolling over on tight turns at high speeds. A bus may overturn if its
inner wheels (those closer to the center of the circular path) lose contact
with the ground. It is fairly easy to solve for the minimum radius of turn, R
min, that a vehicle can undergo at a given speed and not roll over (Let the
normal force under the inner tires go to zero, and set the angular acceleration
of the vehicle to zero as well). Let us also assume that the sharp turn
is not banked. We then find:
R min = (v2h)/(gx)
Here, g is the surface gravity of the Earth, h is
the height of the center of mass of the vehicle off of the road, and x is
the radial distance from the outer wheels to the center of mass of the vehicle.
If the bus in the film was travelling at 50 mph (its minimum allowable),
then the minimum allowable radius turn it could safely make would be 50(h/x)
in units of meters. It is for this reason that Reeves' character rightly
instructs every person on the bus to go to the right side of the bus before
attempting the sharp right turn up ahead. This shifts the center of mass
of the entire vehicle (passengers included) from the center to slightly
off-center, thereby increasing x by a small factor.
How big an effect could this shift in mass have had on their survival?
Well, if the bus were 5000 kg and contained 15 people, then the total
mass of the bus and its contents would be about 6000 kg. If 1000 kg of
that total mass (the people) were to all pile up on the extreme right side,
then the center of mass would shift outward from the center by a factor of one
sixth. If the total width of the bus were 2 meters, then this mass shift
would allow x to grow from 1 m to 1.17 m.
By examining the R min equation, we see that the action
hero's intuition to shift the people to the right side brought the limiting
curve radius down by 17%. If the center of mass of the bus and its
contents was elevated a reasonable 0.75 m off of the ground, then the sharpest turn
the bus could safely undergo at 50 mph without shifting the people
inside would have a radius of 37.5 m. Keanu's stroke of genius
brought this value down to 32 m; the bomb toting bus narrowly averted a roll
over, and the film rolled on at 50 mph.
The Gap Jump
Having survived several dramatic near-death experiences, the passengers
aboard the armed city bus now face their greatest challenge yet: a 50-ft gap in
the overpass ahead. The scene is both "la pièce de resistance"
of the blockbuster film and a gross violation of the laws of mechanics.
When Keanu and friends perform the gravity-defying leap above a
Mario Brothers-like crevice, it is the moment when the film jumps the shark (de
Bont may as well have completed the metaphor by placing hungry sharks in a pool
of water salivating as the bus swoops above their heads).
If we treat the bus as a particle, rather than a rigid body with actual
dimensions, we can approximate the motion of the center of gravity of the bus
using projectile motion kinematics. When a body is launched with an
initial velocity in the horizontal direction (zero pitch angle), it loses
altitude as it sails along horizontally. So, it goes without saying that
no vehicle can jump a gap of any length at any speed unless one or both of the
following conditions are met: (1) the landing is lower than the launch
altitude, (2) the launch velocity points above the horizontal axis by some
angle.
It makes for two nice mechanics problems to solve for the minimum values
associated with each of the two scenarios described above using the data from
the film. Recall that the gap is 50 ft (15.2 m) and note that the bus was
moving at its top speed, 68 mph (30.2 m/s), at the moment of launch.
Using these values, how much altitude would the bus lose if it were to
lift off horizontally? One simple free fall equation shows that it will
lose 4.07 ft (1.24 m). That means that if the landing were at least four
feet lower than the launch altitude, the bus could strike the surface of the
landing rather than bowl it over entirely.
Alternatively, if the landing were at the same altitude as the launch,
what minimum launch angle (road inclination) would allow the bus to reach the
other side? By combining equations of uniform motion along the horizontal
axis and free fall along the vertical axis, we find the minimum angle to be 4.7
degrees. This might not sound like much, but it would be quite noticeable
for a stretch of road.
So, where is the defiance of the laws of physics that I had previously alluded
to? Let us assume that the road was indeed inclined at around five
degrees. And let us pretend that a city bus filled with people could
actually reach 68 mph while driving up such an incline. The real trouble
with this scene is how the pitch of the bus varies during the jump.
If this were a real bus, its pitch would begin to decrease the moment
that the front wheels leave the road. For the brief period of time when
just the rear wheels of the bus are in contact with the road, the vehicle's
pitch would have an angular acceleration in the clockwise (CW) sense.
Then, once the rear wheels have also lost contact with the road, the bus
would have a pitch angular velocity in the clockwise direction. It would
continue to rotate with this angular velocity until it lands (although
'bounces' might be more appropriate). The long term orientation of any
vehicle with two sets of wheels that attempts a jump is a nose dive.
I decided to crunch the numbers on this one, and if you are a mechanics
student, perhaps you would like to try to as well. I assumed that the
center of mass of the bus and its contents was at its center lengthwise, and
found its rotational inertia for pitch by approximating the structure as a thin
rod. I used a bus length of 7 m, and assumed that the two sets of wheels
were 3.5 m apart and that they were spaced symmetrically about the center of
the bus.
With these values and those already mentioned above, I
calculated that the bus would launch with an angular velocity of 0.485 rad/s CW
(due to an angular acceleration of 4.18 rad/s2 CW experienced
while only the rear wheels were in contact with the ground). By the time
the bus reached the other side of the gap, it ought to have rotated through an
angle of 14.2 degrees CW. This means that if the jump were inclined
upwards at 5 degrees, the bus would be inclined downwards at 9.2 degrees when
it reached the other side.
Those familiar with the scene know that this does not occur. As
Bullock and Reeves leave the launch surface, the bus immediately begins to
rotate counter-clockwise. The unjustifiable rigid body
mechanics are so extreme that when the bus lands, its rear wheels strike the
surface first. A student in my class once suggested that maybe the bus
had wings. The truth is that not even wings could cause the rotation
observed in this scene. A more likely justification is that divine
intervention saved Keanu so that he could one day play Neo in The
Matrix.
In summary, the gap jump in Speed is not impossible
from a particle mechanics perspective, but it is impossible from a rigid body
mechanics one. The center of mass of the bus might well reach the other
side of the gap, but the rotation of the bus about this center of mass would
ensure that its nose would strike the road first. The collision dynamics
that would then ensue are very complex, but would likely result in zero
survivors and an ending that no member of the audience saw coming.
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