Sunday, February 12, 2012

"Speed", Featuring Keanu Reeves and Mechanics

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

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.

 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 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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