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.

“Oh crap!!!” is the first thought that runs through your head as you lay on the field/ice/court in excruciating pain immediately following a sports injury.  Whether it is an ankle sprain in soccer, a dislocated shoulder in football, or a broken wrist in basketball, sports-related injuries are painful and frustrating, but unfortunately, inevitable.

The second thought that runs through the injured athlete’s head is, “I wish I could take the last moment back.”  It is an appropriate thought to have, as whenever a collision occurs, the severity of the injury is dependent on just how short the duration of the collision was.  Although impacts in sport between two athletes appear instantaneous, there is actually a small amount of time over which they take place, which we will refer to as dt. 

The extent of the deformation of the athlete’s bone or joint depends on the force imparted during the collision.  If the magnitude of this force is constant throughout the collision, then the total force, F, may be calculated as F = I/dt, where I is the impulse of the collision.  Impulse is the total change in momentum by the bodies involved in the collision.  If a hockey player skates directly into the boards, his change in momentum is his mass multiplied by his speed just before impact.  However, dt depends on the stiffness of the boards.

If the boards were made of concrete, the roster of a hockey team would deplete very quickly, because the dt during impact would be near zero, and the force imparted when crashing into them would be very high.  High-intensity sports lead to high-impulse collisions because the athletes move at high speeds.  Since high-impulse collisions are unavoidable, it is essential that the duration of these collisions be maximized.  The actual hockey boards have ‘give’, and the collision lasts for say one hundredth of a second instead of one thousandth of a second.  The force at the point of collision is then ten times less, and the deformation of the player is as well.

Another example of minimizing the force of impact is when a skier lands a jump.  If the skier does not bend her knees, the impulse will be transferred all at once, and the imparted force through the knee joint will be a maximum value.  By bending her knees, dt is larger, the impulse is ‘absorbed’ gradually, and the lower magnitude force does no harm.  Injury averted.

Once the force applied to an athlete is minimized, it is hopefully distributed over a large area on the athlete.  This is the function of padding.  The extent of the injury is dependent on the pressure transmitted, and p = F/A, where p is pressure and A is area.  The reason a knife is so dangerous is because it cuts even when the force is small, as the area is so tiny.

Most sports injuries occur at joints.  There are two kinds of loads that pass through a joint: forces, which are linear in nature, and torques, which are angular in nature.  It is in fact the torque that is usually responsible for causing damage to a joint.  The torque in the joint is the result of some force being applied some distance away from it.  When an ankle twists the wrong way under an athlete’s own weight, the bending torque is the product of the upwards (reaction) force of the surface and the small distance from the sole of the foot to the joint itself.  A large torque occurs when a large angular impulse is transmitted swiftly.  Again, whether the cause of injury is linear (force) or angular (torque), the level of devastation of the injury is dependent on how quickly it occurs.

I suppose the key to averting sports injuries is to not compete.  I sprained my ankle last week playing soccer, and, I won’t lie: the thought of hanging up my cleats for good did cross my mind as I repeatedly rested, iced, compressed and elevated my injured joint.  Some years ago, a doctor suggested to me that I take up swimming instead of soccer because it is safer.  That may be true, but swimming is also as boring as sin.

I am not willing to give up high-intensity sports, as the benefits outweigh the potential consequences.  If you use the right padding, you minimize the likelihood of injuries occurring.  Still, there are additional measures that can and should be taken for an injury-prone athlete such as myself.  Sports braces for ankles and knees reduce the likelihood of sustaining an injury, and as an engineer, I can appreciate why that is.

When designing a structure, some analysis for an anticipated worst-case environment allows the designer to predict what the maximum force and torque across a given section of the design will be.  If the anticipated load is greater than what a given joint is able to sustain, then the joint must be reinforced to avoid a failure.  A smart way to reinforce a given joint is to give the load a stiffer path to follow.  Stiff braces do this very thing when the joints they are protecting begin to deform.

When I do return to soccer, some weeks from now, I will be sporting an ankle brace.  As a result, the allowable loads for my ankle joint will be higher, and, hopefully, it will not sustain a failure in the near future.  While injuries leave me with much time to write, I much prefer playing sports to writing about them.

Here is a prayer for all athletes out there: “May all of your loads be distributed over a large area, and may your high-impulse collisions be gradual.”

Helicopters do not have a lot going for them.  They are expensive to buy as well as operate, somewhat odd-looking, and, if movies like Die Hard 3 and Cliffhanger are reliable sources of information, they can be brought down with a hand gun or a rock or some rope... pretty much anything nearby.

Helicopters are fairly vulnerable pieces of hardware, but it takes more than a knock in the rotor to drop them out of the sky.

I have traveled by planes, trains and automobiles, but never by helicopter.  From an engineering standpoint, they scare me.  Where possible, it is good to avoid single-point-failure scenarios.  If one airplane engine were to fail, most airplanes can still be flown with the remaining engines.  If the main rotor blade of a helicopter were to fail mid-flight, down it goes.  The main rotor is responsible for the lift force, and without it, it is time to look for a good parachute (and then a good lawyer).

How does a helicopter navigate around?  The principal force is generated by the main rotor, which spins very quickly, displacing vast amounts of air in exchange for lift upwards.  If it wants to maintain altitude, the lift force upwards must be equal to the weight downwards. 

If a helicopter wishes to move forward, it must angle its pitch downwards slightly and spin the rotor a tiny bit faster.  The slightly rotated orientation causes a small component of the lift force to push horizontally.  As some of the lift force no longer pushes directly upwards, the rotor speed must be increased to maintain equilibrium in the vertical direction.

Helicopters are quite inefficient because the majority of the fuel is used not to thrust forward, but to fight gravity.  An airplane’s turboprops are aimed horizontally, and the lift force comes from an offshoot of the plane’s velocity.  What makes an airplane inefficient is the drag force, which is very high because it is proportional to the square of the speed.  Even though planes travel much faster, they are more energy efficient than helicopters.

Helicopter bodies are not all that aerodynamic, which is alright since they tend to move fairly slowly.  However, wind thrusts affect the aerodynamics of the main rotor (one side of the rotor has a different relative speed to the air than the other), which causes a headache for the pilot.

The small rotor on the tail of the helicopter allows it to rotate about its vertical axis (alter its yaw) to change its direction of travel.  It does this by applying a moment (torque) about the center of mass of the helicopter.

I used to see more helicopters in the air.  They used to be helpful to get a picture of the traffic, although you could never decipher a word of information from the reporter due to the extremely loud background noise.  Today, cameras placed in convenient locations accomplish the same goal while saving a great deal of energy and money.

There are some helicopters in use for emergency health situations, and others used to find a missing person.  The vast majority of helicopters are used by military forces at three million dollars a pop. 

The main advantage of a helicopter is that it requires no streets, like an airplane, but also does not require a runway for take-off or landing. 

Helicopters are also unique in that they can perform “station-keeping”.  A subsonic airplane cannot hover; the moment it stops moving forward, its lift force disappears and it plummets downwards.  An airplane is like a shark, which must remain in perpetual motion, even while it sleeps, as no oxygen can be absorbed by its gills if it remains static.  Helicopters are more like fish: agile, but vulnerable.

For all of the agility of helicopters, they tend to be portrayed as targets in the movies, and Rambo has taken down more than a few.  In fact, I challenge you to name a movie that involves helicopters, but does not show one exploding.

Yes, helicopters are fragile creatures, but I roll my eyes whenever an action hero attempts to take one down with dental floss.