Fatal impact – the physics of speeding cars

Key text

Using data from actual road crashes, scientists at the University of Adelaide estimated the relative risk of a car becoming involved in a casualty crash – a car crash in which people are killed or hospitalised – for cars travelling at or above 60 kilometres/hour. They found that the risk doubled for every 5 kilometres/hour above 60 kilometres/hour. Thus, a car travelling at 65 kilometres/hour was twice as likely to be involved in a casualty crash as one travelling at 60. For a car travelling at 70 kilometres/hour, the risk increased fourfold. For speeds below 60 kilometres/hour the likelihood of a fatal crash can be expected to be correspondingly reduced.

Reaction time

One reason for this increased risk is reaction time – the time it takes between a person perceiving a danger and reacting to it. Consider this example. Two cars of equal weight and braking ability are travelling along the same road. Car 1, travelling at 65 kilometres/hour, is overtaking Car 2, which is travelling at 60 kilometres/hour. A child on a bicycle – let's call him Sam – emerges from a driveway just as the two cars are side-by-side. The drivers both see the child at the same time and both take 1.5 seconds before they fully apply the brakes. In those few moments, Car 1 travels 27.1 metres and Car 2 travels 25.0 metres. The difference of 2.1 metres might seem relatively small, but combined with other factors it could mean the difference between life and death for Sam.

The figure of 1.5 seconds is the reaction time of average drivers. A driver who is distracted (eg, listening to loud music, using a mobile phone or has drunk alcohol) may take as long as 3 seconds to react.

Braking distance

The braking distance (the distance a car travels before stopping when the brakes are applied) depends on a number of variables. For example, the slope or grade of the roadway is important – a car will stop more quickly if it is going uphill because gravity will help. The frictional resistance between the road and the car's tyres is also important – a car with new tyres on a dry road will be less likely to skid and will stop more quickly than one with worn tyres on a wet road. If slope and frictional resistance are equal, the factor that has most influence on braking distance is initial speed.

The formula used to calculate braking distance can be derived from a general equation of physics:

where V_f is the final velocity, V₀ is the initial velocity, a is the rate of deceleration and d is the distance travelled during deceleration. Since we know that V_f will be zero when the car has stopped, this equation can be re-written as:

From this we can see that braking distance is proportional to the square of the speed – which means that it increases considerably as speed increases. If we assume that a is 10 metres per second per second and assume that the road is flat and the braking systems of the two cars are equally effective, we can now calculate braking distance for cars 1 and 2 in our example. For car 1, d = 16.3 metres, while for Car 2, d = 13.9 metres.

Adding reaction distance to braking distance, the stopping distance for Car 1 is 27.1 + 16.3 = 43.4 metres. For Car 2, stopping distance is 25 + 13.9 = 38.9 metres. Car 1 therefore takes 4.5 more metres to stop than Car 2, a 12 per cent increase.

We can now see why Car 1 is more likely than Car 2 to hit Sam. If Sam is 40 metres from the cars when the drivers see him, Car 2 will stop just in time. Car 1, though, will plough straight into him. By re-writing the first equation, we can calculate the speed at which the collision occurs:

(where d = 40 metres minus the reaction distance of 27.1 metres = 12.9 metres).

Thus, the impact occurs at about 30 kilometres/hour, probably fast enough to kill Sam. If the car's initial speed was 70 kilometres/hour, the impact velocity would be 45 kilometres/hour, more than fast enough to kill.

These calculations assume that the driver has an average reaction time. If the driver is distracted and has a longer than average reaction time, then he or she may hit Sam without having applied the brakes at all.

Impact on a pedestrian

Because the pedestrian, Sam, is so much lighter than the car, he has little effect upon its speed. The car, however, very rapidly increases Sam's speed from zero to the impact speed of the vehicle. The time taken for this is about the time it takes for the car to travel a distance equal to Sam's thickness – about 20 centimetres. The impact speed of Car 1 in our example is about 8.1 metres per second, so the impact lasts only about 0.025 seconds. Sam must be accelerated at a rate of about 320 metres per second per second during this short time. If Sam weighs 50 kilograms, then the force required is the product of his mass and his acceleration – about 16,000 newtons or about 1.6 tonnes weight.

Since the impact force on Sam depends on the impact speed divided by the impact time, it increases as the square of the impact speed. The impact speed, as we have seen above, increases rapidly as the travel speed increases, because the brakes are unable to bring the car to a stop in time.

Once a pedestrian has been hit by a car, the probability of serious injury or death depends strongly on the impact speed. Reducing the impact speed from 60 to 50 kilometres/hour almost halves the likelihood of death, but has relatively little influence on the likelihood of injury, which remains close to 100 per cent. Reducing the speed to 40 kilometres/hour, as in school zones, reduces the likelihood of death by a factor of 4 compared with 60 kilometres/hour, and of course the likelihood of an impact is also dramatically reduced.

Modern cars with low streamlined bonnets are more pedestrian-friendly than upright designs, such as those found in 4-wheel drive vehicles, since the pedestrian is thrown upwards towards the windscreen with a corresponding slowing of the impact. Cars with bullbars are particularly unfriendly to pedestrians and to other vehicles, since they are designed to protect their own occupants with little regard to others.

Impact on a large object

If, instead of hitting a pedestrian, the car hits a tree, a brick wall, or some other heavy object, then the car's energy of motion (kinetic energy) is all dissipated when the car body is bent and smashed. Since the kinetic energy (E) is given by

it increases as the square of the impact velocity. Driving a very heavy vehicle does not lessen the effect of the impact much because, although there is more metal to absorb the impact energy, there is also more energy to be absorbed.

Less control

At higher speeds cars become more difficult to manoeuvre, a fact partly explained by Newton's First Law of Motion. This states that if the net force acting on an object is zero then the object will either remain at rest or continue to move in a straight line with no change in speed. This resistance of an object to changing its state of rest or motion is called inertia. It is inertia that will keep you moving when the car you are in comes to a sudden stop (unless you are restrained by a seatbelt).

To counteract inertia when navigating a bend in the road we need to apply a force – which we do by turning the steering wheel to change the direction of the tyres. This makes the car deviate from the straight line in which it is travelling and go round the bend. The force between the tyres and the road increases with increasing speed and with the sharpness of the turn (Force = mass × velocity squared, divided by the radius of the turn), increasing the likelihood of an uncontrolled skid. High speed also increases the potential for driver error caused by over- or under-steering (turning the steering wheel too far, thereby 'cutting the corner', or not far enough, so that the car hits the outside shoulder of the road).

Killer speed

All these factors show that the risk of being involved in a casualty crash increases dramatically with increasing speed. In the University of Adelaide study referred to earlier, this was certainly true in zones where the speed limit was 60 kilometres/hour: the risk doubled with every 5 kilometres/hour above the speed limit. A corresponding decrease is to be expected in zones with lower speed limits.

Is the risk worth it? In our hypothetical case, the driver of Car 2, travelling at the speed limit, would have had a nasty scare, but nothing more. The driver of Car 1, driving just 5 kilometres/hour above the limit, would not be so lucky: whether Sam had lived or died, the driver would face legal proceedings, a possible jail sentence, and a whole lifetime of guilt.

Activities

• Exploratorium (USA)
  • Tired weight – calculate the weight of a car by using air pressure.
  • Downhill race – show that acceleration depends on how the mass is distributed.

• Car science (Paul Doherty, USA)
  • Reaction time – measure the distance a ruler falls before you catch it.
  • Accelerometer – measure acceleration with a protractor.
  • Highway seismograph – measure the magnitude of road bumps with a pen and paper.

• On-line Math Applications! (ThinkQuest Junior, USA)
  • Safety – calculate car stopping distances for different speeds (in miles per hour).

• Eberly College of Arts and Sciences (West Virginia University, USA)
  • Driving physics – determine if a car's odometer measures distance or displacement.

• Science upd8 (UK)
  You need to register to access the following (free) activities.
  • Fast cars – student activities on cars including graph interpretation and fill-in-the blanks.
  • High speed smash – students consider balanced and unbalanced forces and their effect on the motion of a drag car.
  • High speed risk – students practice their graph plotting skills and increase their understanding of ethical issues by exploring the risks and benefits of sanctioning high speed driving.
  • Think 30 for a reason – students compare crashes at two speeds and factors affecting vehicle stopping distances.

• 2learn.ca Educators Resource (Canada)
  • Stoplight physics (reaction time applet) – this web activity will reinforce fundamental concepts of velocity, acceleration and frictional force by using the 'reaction time applet' to explore the relationship between the initial velocity and traction of a car and the distance it takes to skid to a stop.

Further reading

Useful sites

Looks at the four components that make up the total stopping distance of a vehicle and includes data showing the comparative stopping distances at different speeds.
http://www.sdt.com.au/safedrive-directory-STOPPINGDISTANCE.htm

Travelling speed and the risk of crash involvement (Road Accident Research Unit, University of Adelaide, Australia)

• Executive summary
  A three-page summary of the study into the effect of travelling speed on the risk of crash involvement at urban speed limits.
  http://raru.adelaide.edu.au/speed/exec.html

• Contents
  A table of contents of all the results of the study and related discussion.
  http://raru.adelaide.edu.au/speed/contents.html

Travelling speed and the risk of crash involvement on rural roads (Centre for Automotive Safety Research, University of Adelaide)

A 2001 report from the University of Adelaide and the Australian Transport Safety Bureau showing that the risk of involvement in a casualty crash increased more than exponentially as speed increased above the mean traffic speed.
http://casr.adelaide.edu.au/ruralspeed/RURALSPEED.PDF

Devices to help drivers control their speed (Michael Paine, Vehicle Design and Research, Australia)

Presents the arguments for speed control devices for vehicles by showing that the number of serious accidents increases as the speed of cars increases. Speed and crashes on the same website presents data showing that serious accidents increase as the speed of a car increases.
http://www1.tpgi.com.au/users/mpaine/Speed.html

Three kinds of friction (Geophysical Institute, University of Alaska, USA)

Describes rolling friction, starting friction and sliding friction and shows how they relate to a car on ice.
http://www.gi.alaska.edu/ScienceForum/ASF8/843.html

National Institute for Advance Transportation Technology (University of Idaho, USA)

A series of topics relating to aspects of safety on the roads.

• Brake reaction time
• Braking distance
• Stopping sight distance
• Decision sight distance
• Passing sight distance

Physics of Racing Series (Brian Beckman, USA)

• Weight transfer
  Explains how inertia and supporting forces give rise to weight transfer through Newton's laws.
  http://www.miata.net/sport/Physics/01-Weight-Transfer.html

• Braking
  Looks at braking in terms of energy conversion (energy of motion into energy of heat of the brakes as well as the interaction between the tyres and the ground).
  http://www.miata.net/sport/Physics/11-Braking.html

Can speeding be justified? (University of Western Sydney, Australia)

Discusses community attitudes to speeding and excuses often given for speeding.
http://www.aggressive.drivers.com/papers/redshaw/redshaw-paper.html

Glossary

kinetic energy. Kinetic energy is the energy associated with a moving object (energy of motion). For more information see Energy basics (Box 3 of Nova: Science in the news topic, Wind power gathers speed).

newton. The SI unit of force. One newton gives an acceleration of 1 metre per second per second to a mass of 1 kilogram. The weight you feel when you hold an average size apple in your hand is about one newton (mass is about 0.1 kilogram and gravitational acceleration of about 10 metres per second per second). It is named after Sir Isaac Newton.