Die Hard

Tough New York cop John McClane (Bruce Willis) is in a pickle. An office building has been taken over by terrorists, everyone else (including his wife) is being held at gunpoint and now they’re threatening to blow up the roof of the building. Mmm, now he’s on the roof of the building and the bad guys are blocking the only exit.

Thinking quickly, Bruce wraps a firehose around his waist and leaps off the side of the building, just as the roof explodes in a mass of broiling flames. It looks great on film, but is this another case of Hollywood Science?

Our Hollywood Sciencebusters Jonathan Hare and Robert Llewllyn did some detective work and found out that the fire hose was indeed strong enough to hold Bruce, but wasn’t very elastic. They claim he would be neatly sliced in two if he really tried to bungee with a fire hose. Yes, yes, we know they did the experiments, but can we prove this theoretically?

The Hollywood Science webteam investigate...

When you free-fall, you speed up because of acceleration due to gravity, and air drag slows you down.

On the Earth, the acceleration rate of a free-falling body is 9.8 metres per second squared. This means that for each second the body is falling, its velocity increases by 9.8 metres per second, up to a limiting velocity of approximately 125 miles per hour (56ms^-1)

So to find out how much damage Bruce will sustain, we first have to know how far he falls.

By looking at the film shot by shot, we estimate that he falls about 35 floors. If we assume each floor is about 3m, then he falls 35 x 3 = 105m

How fast is he travelling when the fire hose runs out? His final speed will be given by his initial speed, added to his acceleration due to the Earth’s pull (gravity), multiplied by the time taken to fall such a distance.

So how long does he take to fall?

The distance travelled can also be worked out by multiplying the speed by the time.If we combine these equations we can work out the time for travelling a set distance with a constant acceleration.

From the time, we can work out the final velocity. If the initial velocity is zero - which it is when he jumps off the top - then his final speed will be acceleration x time.

This is the fastest speed he could possibly reach and in reality will be a little slower due to air resistance. As the velocity of a falling body increases, so does its momentum. Momentum is a physics term; it refers to the quantity of motion that an object has.

Momentum is calculated by taking the mass of a body and multiplying that number by its velocity.

When two things crash, it’s the rate of change of momentum that determines the force (or "wallop") that does the damage.

momentum(p) = mass(m) x velocity(v)

Let’s say Bruce weighs 80 kg, then the momentum he will have at the bottom of his fall will be:

The compressive force (F), experienced as a result of the impact on the Earth can be determined from a physics model using momentum. The force Bruce experiences will be approximately equal to the change in momentum divided by the time it takes him to stop falling.

When Bruce jumps off the building he will first be in free fall until he reaches the end of his hose, the tension in the fire hose will then stop his momentum. A force has been applied that changed his velocity forcing him to slow down and stop.

When Bruce suddenly comes to a halt by the fire hose, he experiences a change in momentum (p), given by p_final - p_initial.

Once Bruce is brought to a halt by the fire hose, he has no momentum.

Therefore,
p_final = 0.

Also, if we assume that he is stopped in the vertical direction and ignore any swinging from side to side, we can use the speed on impact to determine p_initial.

Since the final momentum is 0, the change in momentum p is 0 - 3680 = -3680 kg.m/s. This can also be described as 3680 N.s, where N = 1 kg.m/s squared and shows the force in units of Newtons.

We determine the force of impact using the equation F = p / t.

If we assume it takes 200 milliseconds (0.200 seconds) for Bruce’s fall to come to a stop by the fire hose.

(This is a bit of a guess, as it would probably take a little longer, due to the elasticity of the fire hose and it being 105 metres long.)

We calculate:
F = (3680) / (.200) =18400 N.

This is the force Bruce feels as he is brought to a halt by the fire hose, after falling 35 storeys. So does he have any injuries?

We can work that out too!

This table gives us the tensile strength, in Newtons per metre squared, and the average cross-sectional area in metres squared for four bones vulnerable to injury in this type of fall. By multiplying the tensile strength and the cross section we obtain the maximum force each bone is able to sustain without breaking. Because the fall would extert lateral force on the bones, rather than vertical, the actual forces required to break Bruce’s bones would be smaller.

From the table we see that a force of 18400 N would not break any of the four bones analyzed. Right, so he didn’t get cut in half. But what about head injuries from shear strain? (Shear strain is how the shape of an object changes as the result of forces).

Rapid deceleration of the head can lead to serious injury to the brain stem due to shear strain. This is measured by what’s known as the severity index I , which has been determined experimentally by the equation:

(Man, we love those equations!) Where v is the velocity at impact, t is the duration of impact, and g is the acceleration due to gravity. When the severity index for a collision is above 1000, the collision is fatal. When the value of the severity index is about 400, unconsciousness and mild concussion are the result. So let’s work it out.

Nope! The severity index is 3018, well over the deadly 1000 mark. So it’s curtains for Bruce Willis. In order for Bruce to remain conscious, he would have to come to a halt over a period of 1.8 seconds.

Okay, so how did they do it in the film? The stunt designers have to slow the body’s momentum gradually, to lessen the shear strain. Remember our formula?

The more gradual the change in momentum the weaker the forces acting on the body. For long falls of over 100 feet, they use elastic decelerators, like a bungee cord.

In a bungee jump, the key is the elasticity in the cord.
Although the jumper reaches the end of the cord (L), it continues to stretch (l) until he reaches its full extremity.

Thus the total length of the cord is L+l and the energy has been converted to the elastic potential energy of the cord, acting as a brake.

Since energy is conserved in the jump, the gravitational potential energy of the jumper must equal the elastic potential energy of the cord.