"Speed", Featuring Keanu Reeves and Mechanics

The 1994 action thriller Speed introduced America totwo soon-to-be movie stars in Keanu Reeves and Sandra Bullock.  Thefilm contained all of the escapist elements that a summer outing tothe theatre is supposed to: adrenalin-fuelled intimacy between the twoleads, a smart yet mentally deranged villain, and lots of things thatcan and eventually do go boom. 

Yet, after the credits role, and the movie goers make their wayhome, the images that stick with them are not those of kissing,lunacy, or explosions, but rather the exciting set-pieces involvingmechanics that are continuously on display.  From the elevator on whichthe film opens, to the bus, where the majority of it takes place, to the subwayon 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 wouldsuspect that director Jan de Bont took a physics class as a kid and enjoyed itimmensely.

I suppose it is not surprising that the movie features mechanics,as its title is a key term of kinematics - speed is defined asthe magnitude of velocity.  And, when an ex-cop turnedpsycho attaches a bomb to a city bus, he programs it with this kinematicparameter 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 eachaction sequence, I would pause the film to discuss the key concepts ofmechanics on display, and even solve explicitly for some of the unknownparameters.  As this exercise is quite time-consuming, I simply encouragemy students to try this activity on their own.

In the first scene alone, many aspects of mechanics are highlighted whenan elevator filled with innocent people threatens to plummet to theground.  The periods of free fall experienced by both theelevator and those inside begs several questions, like "Shouldthe passengers float upwards?" and "Would they increase theirlikelihood of survival if they jumped just before the cabin hits theground?"  I'll leave readers to consider these on their own.

When the cabin and its contents are supported by a single rope, how muchtension manifests inside it?  Does the rope extend, and if so, by howmuch?  Why does the supporting crane above break?  I'll answer thislast one: the tension in the cord creates a large moment (or torque) about thesupport structure.  The bending moment leads to a local stress that islarger 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 thisfashion, but instead, I would like to focus on two particular action sequencesin some detail.  These two sequences occur on the fast-moving bus, and Ialways discuss them with my mechanics classes.

 The Sharp Turn

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 followsa circular path, it has an inherent centripetal acceleration that pointstowards the center of the path.  The magnitude of this accelerationis v2/R, where v is the speed of thevehicle and R is the radius of the circular path.  

Typically, when a vehicle rounds a bend, it reduces its speed to reducethe magnitude of this acceleration.  A car does this to avoid skiddingoutwards during the turn.  This would occur if the inertial term of thecar (its mass multiplied by its centripetal acceleration) were to exceed themaximum static friction between the tires and the road.  To help carsavoid skidding, curved roads are often inclined upwards at some angle - theresult of this feature is that the inertial term must overcome both thestatic friction as well as a small component of the car's weight in order forit to skid.

As our friends in Speed preparefor a sharp turn, their concern is not that they will skid, but that they willroll over.  A typical sedan has a center of mass that is quite close tothe 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 ofrolling over on tight turns at high speeds.  A bus may overturn if itsinner wheels (those closer to the center of the circular path) lose contactwith the ground.  It is fairly easy to solve for the minimum radius of turn, Rmin, that a vehicle can undergo at a given speed and not roll over (Let thenormal force under the inner tires go to zero, and set the angular accelerationof the vehicle to zero as well).  Let us also assume that the sharp turnis not banked.  We then find:

R min = (v2h)/(gx)

Here, g is the surface gravity of the Earth, h isthe height of the center of mass of the vehicle off of the road, and x isthe 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 rightlyinstructs every person on the bus to go to the right side of the bus beforeattempting the sharp right turn up ahead.  This shifts the center of massof the entire vehicle (passengers included) from the center to slightlyoff-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 totalmass of the bus and its contents would be about 6000 kg.  If 1000 kg ofthat 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 onesixth.  If the total width of the bus were 2 meters, then this mass shiftwould allow x to grow from 1 m to 1.17 m.  

By examining the R min equation, we see that the actionhero's intuition to shift the people to the right side brought the limitingcurve radius down by 17%.  If the center of mass of the bus and itscontents was elevated a reasonable 0.75 m off of the ground, then the sharpest turnthe bus could safely undergo at 50 mph without shifting the peopleinside would have a radius of 37.5 m.  Keanu's stroke of geniusbrought this value down to 32 m; the bomb toting bus narrowly averted a rollover, and the film rolled on at 50 mph. 

The Gap Jump

Having survived several dramatic near-death experiences, the passengersaboard the armed city bus now face their greatest challenge yet: a 50-ft gap inthe 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 aMario Brothers-like crevice, it is the moment when the film jumps the shark (deBont may as well have completed the metaphor by placing hungry sharks in a poolof water salivating as the bus swoops above their heads).

If we treat the bus as a particle, rather than a rigid body with actualdimensions, we can approximate the motion of the center of gravity of the bususing projectile motion kinematics.  When a body is launched with aninitial velocity in the horizontal direction (zero pitch angle), it losesaltitude as it sails along horizontally.  So, it goes without saying thatno vehicle can jump a gap of any length at any speed unless one or both of thefollowing conditions are met: (1) the landing is lower than the launchaltitude, (2) the launch velocity points above the horizontal axis by someangle.

It makes for two nice mechanics problems to solve for the minimum valuesassociated with each of the two scenarios described above using the data fromthe film.  Recall that the gap is 50 ft (15.2 m) and note that the bus wasmoving 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 tolift off horizontally?  One simple free fall equation shows that it willlose 4.07 ft (1.24 m).  That means that if the landing were at least fourfeet lower than the launch altitude, the bus could strike the surface of thelanding 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 theother side?  By combining equations of uniform motion along the horizontalaxis and free fall along the vertical axis, we find the minimum angle to be 4.7degrees.  This might not sound like much, but it would be quite noticeablefor a stretch of road.

So, where is the defiance of the laws of physics that I had previously alludedto?  Let us assume that the road was indeed inclined at around fivedegrees.  And let us pretend that a city bus filled with people couldactually reach 68 mph while driving up such an incline.  The real troublewith 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 momentthat the front wheels leave the road.  For the brief period of time whenjust the rear wheels of the bus are in contact with the road, the vehicle'spitch would have an angular acceleration in the clockwise (CW) sense. Then, once the rear wheels have also lost contact with the road, the buswould have a pitch angular velocity in the clockwise direction.  It wouldcontinue to rotate with this angular velocity until it lands (although'bounces' might be more appropriate).  The long term orientation of anyvehicle 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 mechanicsstudent, perhaps you would like to try to as well.  I assumed that thecenter of mass of the bus and its contents was at its center lengthwise, andfound its rotational inertia for pitch by approximating the structure as a thinrod.  I used a bus length of 7 m, and assumed that the two sets of wheelswere 3.5 m apart and that they were spaced symmetrically about the center ofthe bus.  

With these values and those already mentioned above, Icalculated 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 experiencedwhile only the rear wheels were in contact with the ground).  By the timethe bus reached the other side of the gap, it ought to have rotated through anangle of 14.2 degrees CW.  This means that if the jump were inclinedupwards at 5 degrees, the bus would be inclined downwards at 9.2 degrees whenit reached the other side.  

Those familiar with the scene know that this does not occur.  AsBullock and Reeves leave the launch surface, the bus immediately begins torotate counter-clockwise.  The unjustifiable rigid bodymechanics are so extreme that when the bus lands, its rear wheels strike thesurface first.  A student in my class once suggested that maybe the bushad wings.  The truth is that not even wings could cause the rotationobserved in this scene.  A more likely justification is that divineintervention saved Keanu so that he could one day play Neo in TheMatrix.

In summary, the gap jump in Speed is not impossiblefrom a particle mechanics perspective, but it is impossible from a rigid bodymechanics one.  The center of mass of the bus might well reach the otherside of the gap, but the rotation of the bus about this center of mass wouldensure that its nose would strike the road first.  The collision dynamicsthat would then ensue are very complex, but would likely result in zerosurvivors and an ending that no member of the audience saw coming.


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