Impulse (Δp) – A central concept in safety engineering

The concept of impulse is one of simplest in all of mechanics. It's just the change in momentum during some process, like a collision:

$$\Delta p = p_{final} - p_{initial}$$

Yet when we relate impulse to force (which we'll do below), we get a powerful tool for understanding the forces on humans and animals in accidents like vehicle crashes and falls. It's a way of thinking that led to important safety features like seatbelts and air-bags in cars, stretchy rock climbing ropes, parts of cars that crumple strategically in a collision, and many other devices that have saved many lives. Aahh ... Science!

The basic idea is that when a living thing, like a human, a dog, a cat, livestock ... is involved in a collision, its momentum can change dramatically over a very short time. It can go from a very high value to zero very rapidly. Imagine your body traveling at 60 mi./h when your car slams into a big tree. In a tenth of a second or less, your body will go from a very high momentum to zero momentum. Bodies can only take so much force, and it turns out that the shorter the time it takes to reduce your momentum to zero, the larger the force on you. If we can do things to increase the time it takes to undergo that momentum change, the force will be reduced and you might survive.

The advertisement on the left was run by the U.S. Department of Transportation to encourage people to use seatbelts, one of many devices now in cars meant to reduce the forces that could kill a person in a car collision.

Impulse and force

The real value of considering Δp is relating it to force. Let's start by remembering that

$$\Delta p = p_{final} - p_{initial}$$

Now think about units. The unit of force is the Newton (1N = 1Kg·m·s-2), and if we multiply by time (in seconds), one of those seconds in the denominator divides out, and we get the units of momentum, Kg·m·s-1

So force multiplied by the change in time is just the change in momentum, and we can rearrange to find a new formula for force:

$$F \Delta t = \Delta p \: \: \longrightarrow \: \: F = \frac{\Delta p}{\Delta t}$$

If we break momentum down into velocity and mass, we can further refine our formula:

$$F = \frac{\Delta p}{\Delta t} = \frac{\Delta (mv)}{\Delta t} = \frac{m \Delta v}{\Delta t}$$

Now this is really a remarkable result. It says that for a given momentum change, or equivalently for a given velocity change of some fixed mass – like a person, the force is inversely proportional to the amount of time it takes to undergo that momentum or velocity change.

The longer it takes to change momentum, the less force is applied or experienced. The shorter the time, the more force. Here's a graph of the situation.

Impulse and Force

Impulse is the change in momentum during some process, like a collision:

Δp = pfinal - pinitial

Force and impulse are related:

Force is inversely proportional to the time it takes to change momentum.

Why does this matter: Seatbelts and airbags

One of the most dramatic momentum changes that can be undergone by a human body is slamming into the steering wheel (or worse, being ejected from the vehicle through the windshield) during a high-speed crash. Air bags and seatbelts have dramatically reduced the death rate from such crashes.

Consider the crash-test dummy below, pictured in a test of an air bag. Let's say that a person is traveling at 60 mi./h and hits an immovable object.

First we'll convert 60 mi./h to meters/second:

$$ \begin{align} \left( \frac{60 \, mi}{h} \right)\left( \frac{1 \: h}{3600 \, s} \right)&\left( \frac{1609 \, m}{1 \, mi} \right) \\ &= \: 26.82 \: m/s \end{align}$$

Now let's say the average person has a mass of 60 Kg, but we're really only talking about the movement of the torso of a seat-belted driver, so we'll say that the mass is 30 Kg. Then the initial momentum is

$$ \begin{align} p+i &= (30 \, Kg)(26.82 \, m/s) \\ \\ &= \: 805 \, Kg \cdot m/s \end{align}$$

Because the final momentum is zero, the impulse is:

$$\Delta p = 805 \, Kg \cdot m/s$$

(By definition, this should be negative, but we'll just redefine our coordinate system to make a loss of momentum positive – no harm in that.) Now let's say that our driver's head is 30 cm (0.3 m) from the steering wheel. At 26.8 m/s, the time it would take for the head to contact the wheel is

$$\Delta t = 0.3 \, m \left( \frac{1 \, s}{26.82 \, m} \right) = 0.0112 \, s$$

That's 11.2 milliseconds (ms) – pretty fast. With that time, we can calculate the force felt by the head as it comes to a stop against the wheel.

Now 72 KN is a lot of force. In fact, most human bones break at a force below 10 KN.

What if we could increase the time to the steering wheel by, say, a factor of 10. Instead of taking 0.01 s to stop, let's say it took 0.1 s. Now the force would be reduced to:

$$F = 7.2 \, KN = 7200 \, N$$

which is in the range of survivable forces. That's just what an airbag is designed to do: It increases the time it takes to undergo the impulse change.

Airbags work by sensing rapid deceleration of the vehicle, pressurizing the bag explosively when needed, then deflating more slowly, say over several tenths of a second.

Seatbelts have a dual effect. They are designed to keep car occupants inside the car where it's likely to be safer during a crash, and they stretch just a bit, helping to increase the time it takes for the impulse to be absorbed.

Cars are full of force-reducing measures

The drawing above shows some of the safety features built into most modern vehicles. The driver is surrounded by the passenger "cage," a steel framework designed to maintain a shell of saftey around the passengers.

Inside that cage are the safety restraints – seatbelts, and several airbags, some even on the sides of the car in case of a side impact.

In back and front are "crumple zones," regions of the car that are designed to collapse during a collision, not just transfer energy to the occupants. These zones further lengthen the time it takes to slow down and lose momentum.

In the front of passenger cars, the engine is often engineered to drop below the passenger compartment so that it doesn't intrude into it.

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