Insultingly Stupid Movie Physics

by Tom Rogers

  When the bus lands, it has the same downward vertical velocity as it would have if it were raised using a crane and dropped from the same height as the top of the trajectory. If the bus lands on a horizontal surface, it will slam into it with a considerable force. The bus’s downward velocity component will drop almost instantaneously to zero when the bus lands on such a surface, yielding extremely high accelerations and, subsequently, extremely high forces (see Chapter 10).

  Here’s the big surprise: as long as the bus reaches the correct takeoff speed at the correct takeoff angle (assuming negligible air resistance and vehicle rotation), the bus’s mass is not a factor in the length of the jump! Why? It goes back to Galileo, who was perhaps the first person to understand that all objects fall at the same rate regardless of their mass. Aside from takeoff speed and angle, the rate of falling is the primary factor determining the distance of the jump.

  Although a jump with a takeoff and landing ramp would have been more realistic than the one filmed for the movie, the doubleramp jump also has its hazards, not to mention problems with reality. If the bus takeoff velocity were too low, the front of the bus would smash into the landing ramp. Too fast and the bus would partially overshoot the ramp and experience a hard landing, be undrivable, and explode or merely break the bus driver’s back and seriously injure most of the passengers. Realistically, the speed would have to be higher than the exact level for the crossing because it’s better to risk a hard landing than a crash into the landing ramp.Throw in the need to keep the bus drivable and the margin for error is next to nothing.

  But all this discussion about adjusting to the precise speed needed for surviving the jump is hypothetical. Takeoff and landing ramps that are exactly the correct angle for making the jump are not likely to be found on overpass bridges. For one thing, changing from an upward to an equal but downward slope in only 50 feet of distance would cause vehicles traveling above sixty miles per hour to go airborne as they crossed the completed bridge’s peak—a poor design at best. So while the jump may be possible, it’s pretty far-fetched.

  Still, the true bridge-jump believers are right, in a sense. In theory, the bus jump could have been made even without the ramps—that is, if the bus had been driven fast enough to put it in a circular orbit with a radius equal to the radius of Earth. This jump would have required a takeoff velocity of about 17,700 miles per hour (28,500 kph)—a little quick for most city busses. Air resistance would also have been a factor but might not have been all that bad for a mere 50 feet. The driver and passengers would have likely blacked out from the acceleration required for reaching 17,700 miles per hour, if they survived it. The sonic boom would have smashed windows and rattled nearby buildings but, hey, it would certainly have added excitement. So, in the next movie bus jump, maybe a nuclear rocket scientist will be aboard and just happen to have his latest miniature nuclear rocket creation in his brief case.

  JUMPING HULKS

  There’s no question that the Incredible Hulk is one bad dude and, at first glance, the jumps attributed to him in his movie seem reasonable. But they’re not. Such a jump yields projectile motion similar to a bus jump off a ramp. Once airborne the only forces acting on the projectile—in this case the Hulk—are air resistance and gravity, neither of which can help make the jump longer. Sometimes aerodynamically shaped objects such as the discus used in Olympic events can experience lift, a factor that does extend the distance traveled. But lift is unlikely with a boxy object such as the Hulk. This lack of an assisting force after takeoff means that the length of the jump will be dictated entirely by the take off velocity and angle. The movie depicts the Hulk making jumps on the order of a kilometer—an impossibility given his rather slow takeoff velocity.

  HOW JUMPING DISTANCE SCALES UP IN CRITTERS

  Equation 9.1 established that the length of a jump is only a function of the takeoff velocity squared and the angle. The same relationship is true for cars, critters, or people. Assume that a critter starts its jump from zero velocity and reaches takeoff velocity by the time its feet (or paws) leave the ground. From kinematics

  Where:

  v = takeoff velocity

  a = constant acceleration in the same dimension as v m = mass of the critter being accelerated

  d = distance the leg force acts on the critter before the critter’s feet leave the ground

  combined with F = ma yields:

  Where:

  F = the constant force provided by legs

  but

  F is proportional to the cross sectional area A of the muscle producing the force. So, again, referring to equation 9.1 for calculating the range or horizontal length of a jump:

  The range X is proportional to V2 or (A/m)D

  A scales up with the square of the scaling factor (S). m scales up with cube of the scaling factor (S3) and D with S (see Chapter 4). If an animal is scaled up by the factor S, the new jumping distance will be

  In other words, the new jumping distance will be the same as the old, assuming that the animal was not scaled up so much that it collapsed under its own weight.

  So, why are tall people often able to jump higher than short ones? They are not scaled up proportionally. Generally, the big difference between short and tall people is the length of their legs. Leg length makes up a larger proportion of a tall person’s height as compared to a short person. Proportionately longer legs would be an advantage for jumping because the jumping force they produce when bent and straightened rapidly during a jump would be applied over a longer distance. However, the torso moved by the legs would still weigh about the same as a short person’s torso. Although the tall person’s legs would weigh more, the torso still accounts for most of a person’s weight. Hence a tall person could jump farther than a short one, assuming both had similar athletic conditioning and skill.

  So what takeoff velocity would the big guy need to travel 1 kilometer in a single leap, and what would he look like making such a leap? If we ignore air resistance and assume the takeoff angle is forty-five degrees, the Hulk would need a takeoff velocity of about 222 miles per hour (357 kph). He would appear to move away quickly then seem to be slowing down as his image got smaller with increasing distance. While the movie is not a perfect match for the calculated behavior, it’s at least in the ballpark, except for one very sticky detail: air resistance cannot realistically be ignored. The Hulk is too large, too boxy, and too fast.

  Accounting for air resistance is tricky. First, air resistance changes with velocity—unlike a more cooperative force such as gravity, which remains constant (at least in projectile-motion problems). But it gets worse: at low speeds air resistance can be approximated as follows:

  (air resistance) = (coefficient of drag) × (cross-sectional area) × (velocity)

  At higher speeds the velocity term in the above equation changes to velocity squared. In other words, the air resistance becomes even more dependent on velocity.

  As for the coefficient of drag (CD), it’s just a constant, or as engineers call it, a fudge factor—a factor tossed into the equation to fudge the numbers so they come out right. And where would this fancy fudge factor come from? From wind tunnel measurements on the Hulk. That poses a problem: the Hulk is not on hand and probably wouldn’t cooperate if he were. To complicate matters further, the CD measurement is only good for one wind direction and one Hulk configuration. When jumping, the Hulk would have to always keep himself oriented in the direction of his velocity and hold his arms and legs in the same position as when his CD was measured. Otherwise, his CD, not to mention his cross-sectional area would change during his jump.

  Without turning the analysis into a career, about the best that can be done is to model the Hulk as a sphere and use a computer simulation package such as Interactive Physics to get an idea of how much air resistance affects the Hulk’s jumping distance. When we do so, we get an amazing result: the Hulk’s takeoff speed must be around 1,250 miles per hour (2,020 kph), faster than the speed of sound, faster even tha
n a speeding .22-caliber long rifle (LR) bullet (873 mph or 1,410 km/hr). Okay, we made a lot of assumptions, but keep in mind that the maximum distance a .22-caliber bullet will travel is only about 1.1 miles (1.8 km). Since the bullet is far more aerodynamic than the Hulk, the Hulk would need to start at a much higher speed to reach even the shorter distance of 1 kilometer (0.62 mi).

  So how would the Hulk really look if he made a 1-kilometer jump? He’d look like a giant green cannon ball; he’d go so fast he’d be a blur. His landing velocity (134 mph, or 216 kph) would be far slower than his takeoff velocity due to the effects of air resistance, but when he landed the impact would be impressive.

  The Hulk’s takeoff would also be dramatic. If he were standing still and decided to leap, he would first bend his knees then very quickly push off. As he straightened to his full height, his feet would break contact with the ground and he’d be launched into the air. The acceleration propelling him into the air would only occur during the short distance between his bent knee and fully straightened position: a distance of, at most, 30 inches (0.8 m). Once his feet broke contact with the ground, he would no longer be able to increase his takeoff velocity. His average acceleration during takeoff would exceed 30,000 gs. The force his feet exerted on the ground would be his normal weight times the acceleration in gs, or about 15,000 tons—enough to break concrete.

  Pound for pound, the Hulk’s muscles would have to be thousands of times stronger than human muscles to make his lengthy jumps. If an animal is scaled up or down, the distance it can jump (assuming it does not collapse under its own weight) will not change. For example, if a flea can jump 20 inches (0.5 m) in its normal size it will still only be able to jump about 20 inches if scaled up to the size of a cricket. The only way it could jump further would be to get stronger muscles. The Hulk is similar in build to a Neanderthal wrestler on steroids (had there been one). Scale up such a wrestler to the Hulk’s size, and he’d still only be able to jump his normal amount—a few meters—not the Hulk’s incredible 1,000-meter leap.

  So when it comes to jumps in Hollywood movies, the important question is not what the laws of physics say; it’s what the director says. When the director says jump, rather than wasting time on calculations, moviemakers simply ask how far and how high.

  Summary of Movie Physics Rating Rubrics

  The following is a summary of the key points discussed in this chapter that affect a movie’s physics quality rating. These are ranked according to the seriousness of the problem. Minuses [–] rank from 1 to 3, 3 being the worst. However, when a movie gets something right that sets it apart, it gets the equivalent of a get-out-of-jail-free card. These are ranked with pluses [+] from 1 to 3, 3 being the best.

  [–][–] Creatures making incredibly long jumps at low velocities with little indication of high takeoff or landing force.

  [–][–] Hapless souls, heroes or otherwise, cast through windshields for no good reason.

  [–] Large creatures with leaping abilities far greater than the smaller versions they were scaled up from.

  [–] Simulated vehicle jumps that depart from reality.

  [+] Vehicle jumps filmed under realistic conditions.

  CHAPTER 10

  ACCELERATION AND EWTON’S SECOND LAW:

  How to Get Started, Use the Brakes, or Change Direction, Hollywood Style

  NEWTON’S SECOND LAW—A SYNOPSIS

  Newton’s second law rests on the definition of acceleration, which like most things in physics doesn’t have the same meaning as in everyday language. Like force and velocity, acceleration is one of those strange quantities called vectors—represented by arrows. The arrow indicates the quantity’s direction, and the length of the arrow indicates the quantity’s magnitude or size. Acceleration is simply a measure of how fast the velocity is changing. A change in velocity can mean that an object is slowing down, speeding up, or changing direction. If the change happens quickly, we get a high acceleration; and if it happens slowly, we get a low acceleration.

  Newton’s second law teaches that

  Force = (mass) (acceleration)

  or

  F = ma (EQUATION 10.1)

  In other words, force and acceleration are directly proportional. They go hand in hand. If one increases, the other must also increase; if one decreases, the other must also decrease. The arrow representing acceleration and the arrow representing force always point in the same direction. By contrast, the arrow representing velocity can go in an entirely different direction from the arrows representing the force or acceleration.

  If the arrow representing acceleration or force points in the opposite direction from the arrow representing velocity, the object is slowing down. If the two arrows point in the same direction, the object is speeding up. For physics purists, the term deceleration—gasp!—doesn’t exist. Okay, some physics books regrettably use this abominable term, but to the pure of heart, it’s bad form.

  To many, it seems like Newton’s second law is just a repeat of Newton’s first law in equation form. There’s some truth in that, but there’s an additional difference: a complete description of Newton’s first law defines something called an inertial frame of reference, which must exist for Newton’s second law to be true. A frame of reference is whatever is assumed to be stationary (often the floor). An inertial frame of reference is one where Newton’s first law holds true.

  Is there a place where Newton’s first law doesn’t hold true? Yes, and it’s found even in some fairly ordinary places. An inertial frame of reference can be moving at constant velocity but cannot be accelerating with respect to any other inertial frame of reference. An inertial frame of reference cannot, for example, be a merry-go-round because its parts are changing direction or accelerating with respect to the ground as they go around and around. If a person riding on one side of a merry-go-round tries to throw a ball straight across to a person riding on the other side, the ball will appear to go in a curved path relative to the rotating merry-go-round. Even Earth’s surface cannot be considered a true inertial frame of reference because it’s rotating. But don’t worry; Earth is so large that it can be considered as though it were a flat immobile surface, at least for all the examples in this chapter.

  HOW TO ENJOY A CRUISE SHIP CRASH

  A runaway cruise ship (Speed II [PGP-13])—its engines unstoppable—is headed straight for a dock in the greatest movie ship crash scene ever filmed (not that ship crashes are common). Its passengers scream as the boat rips through the wooden dock. Conveniently, the first mate calls out the boat’s speed to reassure viewers that the boat is indeed slowing down (duh). In the middle of the crash, for added excitement (as if there weren’t enough), the movie’s heroes are hurled through the boat’s windshield onto the deck below. The ship’s windshield—made of laminated glass to resist wave impact during storms—would lacerate the flesh, shatter the face bones, and knock out the teeth of any unfortunate soul who smashed through it. But miraculously the heroes are uninjured. After taking out the dock and a few condos, the giant boat comes to rest.

  So, why were the people aboard the craft screaming? They should have relaxed in a deck chair, sipped a cold drink, and enjoyed the spectacle. There never was any danger, at least not to them. As for the heroes crashing through the ship’s windshield, consider the same situation for a car traveling forty-five miles per hour (72 kph) that hits a brick wall and stops almost instantaneously, in say 0.01 seconds. The driver (who considers seat belts unmanly) crashes through the windshield. By contrast, the boat is traveling all of seven miles per hour (11 kph) and takes thirtythree seconds to stop. Common sense alone says that the boat crash is incredibly gentle compared to the car wreck.

  A quick calculation shows that the stopping acceleration is over 200 gs for the car and around 0.01 gs for the ship. People will stay put in the car or boat if they have exactly the same acceleration as their respective vehicle. If not, the people will continue moving forward as the car or boat stops. We say that they are “thrown” fo
rward, but really they’re not. They passively continue moving forward until a force or combination of forces acts to stop them: in the case of the car’s driver, crashing through the windshield and smashing into the wall beyond it provides the combination of restraining forces. In the more gentle case of the ship, the people will never move forward with respect to the ship as long as the friction force between the passengers and deck is high enough to restrain them.

  The maximum friction force acting on passengers standing on the deck would typically be at least half their weight. Such a force could keep a passenger fixed to the deck with ship stopping accelerations up to 0.5 gs, or about fifty times greater than depicted in the movie. While it might be difficult to keep one’s balance under such conditions, as soon as one fell to the deck, the friction force would be restored and any forward motion with respect to the deck would cease.

  The windshield of a ship would be capable of providing a very high stopping force. For the windshield to break from human impact, the ship would need a stopping acceleration of at least five gs. At a speed of seven miles per hour (11 kph), the ship would have to slam into a perfectly solid barrier and come to a complete stop in a remarkably small distance of 3.8 inches (9.6 cm). This distance includes any deformation or crumpling of the ship’s front.