Or, “How simple physics can keep you from killing somebody.”
The Problem
You are sitting in stand-still traffic in the fast lane of a commuter highway. You are impatient. You have an inflated sense of your own self-importance to the human race. Your job is so important to the fate of our species that if you don’t get to work on time the world might end. You make a decision. You are going to jump from the stand-still fast lane to your left, across a line of plastic barricades and into the currently empty high-occupancy vehicle (HOV) lane. You check your left mirror and see no cars in the lane within your field of view.
Not only are you breaking the rules of HOV lane usage (it is a violation to cross the barricade to enter or leave the HOV lane), you are dangerously ignorant of basic physics to the point where you are potentially about to kill or maim an individual.
The Physics – Stopping Distance
A car moving in the HOV lane at the maximum allowed speed (60 mph) requires a minimum distance to come to a complete stop. Let’s explore this problem with physics.
Let us begin with the kinetic energy (KE) of the car. This is given simply by the following equation:
KE = (1/2)mv2
where m is the mass of the car (in kilograms) and v is the speed (in meters per second). We need to convert the speed in miles per hour (mph) to meters per second:
60 mph = (60 mph) x (1/3600 h/s) x (1594.2 meters/mile) = 26.57 m/s
The mass of a 2-ton car, in kilograms, is 1814.36 kg. Thus a 2-ton car moving at 60 mph has the following kinetic energy, in Joules:
6.404×105 J
This is the amount of energy that needs to be taken out of the car in order to stop it (bring its speed to zero meters per second). When you hit the brakes on a car, you are applying a friction force to the wheels to slow them. The wheels are still in motion, so they are working against kinetic friction from the brakes. Friction is a contact force, so depending on the contact area the force will be lesser or greater.
Let’s look at one specific case where the braking distance is made as short as possible. This is when the driver who must stop hits the brakes and locks them so that the wheels stop turning. The static friction that normally holds a point of contact between the wheels and the road now turns into kinetic friction as the wheels skid on the road. We need to know the coefficient of kinetic friction between rubber tires and a highway surface. In Dallas, this is concrete. Let’s assume dry conditions. The number is μk=1.0 [1]. The force due to kinetic friction between the car and roadway is given by:
Fk = μk(mg)
where m is the mass of the car and g is the acceleration due to gravity at the surface of the Earth (9.8 m/s2). The force due to friction, in Newtons, is then 1.778×104 N.
How do we determine the stopping distance? A force like friction does WORK – that is, exerts itself in a direction opposite the motion along a certain distance – until all the kinetic energy of the car is gone and there is no more movement. All of the energy of the car is then expended overcoming the opposing force of friction. The equation that relates the two is:
(1/2)mv2 = μk(mg) D
where D is the stopping distance (the distance over which friction acts against the motion of the car). So, what is D for a 2-ton car moving originally at 60 mph with rubber tires sliding on a dry concrete roadway?
D = 36.02m = 118.18 feet.
From this we conclude that if you, the driver who wants to jump into the HOV lane (which is already breaking a law), see a car in the HOV lane that is within 120 feet of your vehicle, then jumping into the HOV lane anyway is an intentional act of manslaughter, suicide, or both. If you are a terrible judge of distance, which you will be because those side mirrors distort distances (we’ll get to that in a bit), you should just stay where you are.
Please note! The above calculation assumed a dry day. If there is anything wet or slick on the road (water, oil, etc), this reduces the coefficient of friction. On a wet day, μk=0.3, which makes the stopping distance D under those conditions 3 TIMES LONGER. So instead of 118 feet, the car requires 354 feet to come to a complete stop. That’s almost a twentieth of a mile . . . !
Also, note that the mass of the car plays no role in the stopping distance – it cancels from both sides of the equation. So a Chevy Equinox, which weighs about 63% more than my Honda Civic, has the same 118 foot stopping distance.
Speed plays a huge role in the stopping distance. If a car is speeding in the HOV lane – going 70 mph, for instance, which is 17% faster – then since the stopping distance goes as the SQUARE of the speed the distance increases not by 17% but by 36% over the car at 60 mph. So the stopping distance at 70 mph is 161 feet!
OK, but let’s say you’ve looked in that left-side rearview mirror and see no cars in the HOV lane. Are you safe?
Human Reaction Time and the Speed of the Other Vehicle
Human reaction time is about 0.2 seconds. That means from the moment you decide to take an action to the time you physically initiate the action is about 0.2 seconds. So once you commit to illegally jumping into the HOV lane, it will be about 0.2 seconds before your foot presses on the gas and your hands turn the wheel to cut into the lane. What can happen in 0.2 seconds?
That depends on the speed of the vehicle you didn’t notice before you decided to make your move. How far can a car travel when at a speed of 60 mph?
That one’s easy. The relationship between distance, time, and speed is:
D = vt
so if the car in the HOV lane is going 60 mph (26.57 m/s) then in 0.2s it can travel 5.3 meters, or about 17 feet.
That doesn’t sound so bad, but what if the HOV lane in your mirror only sees back as far as the crest of a hill? By the time you start making your move, the car in the HOV lane will have moved 17 feet, which could easily bring it over the crest of the hill and into view. But now you, the driver jumping into the HOV lane, are not looking for a car anymore; you’re too busy spinning your steering wheel and hitting the gas to pull into the HOV lane.
You will accelerate into the HOV lane. How long will it take for you to cross the distance from the fast lane to the HOV lane? That’s a distance of about 1 car length, which for a full-size car is 5 meters. If you floor the accelerator on a typical car, how long will it take to travel the 5 meters into the HOV lane? Remember, we’re adding this to your reaction time, which was the time between when you decided to jump and when you physically began the steps needed to jump. So whatever we find here, we add 0.2 seconds (17 feet traveled by the car you didn’t see in the HOV lane).
A typical car has a maximum rate of acceleration of about 0.5g, or one-half the acceleration due to gravity at the surface of the Earth [2]. The relationship between time, acceleration, and distance is:
D=(1/2)at2
if one assumes that the acceleration begins from rest (v=0 at t=0). So how much time does it take to go the 5 meters into the HOV lane?
Solving for time, we find t = 1.4 seconds! Add to that your reaction time – 0.2s – and the whole maneuver (from deciding to go to making it into the lane) takes 1.6 seconds! And that is a best case scenario for a standard car – I assumed the upper range of acceleration for a full-size car.
So you’ve jumped into the lane. It took 1.6 seconds for the whole move. How far does a car traveling 60 mph in the HOV travel in that time? The answer is 136 feet. Holy crap! That’s more than the braking distance required for that same car. In the time it took you to execute a bad and illegal decision, any car already in the HOV lane might not stand a chance since it will travel further than braking distance in the time it takes you to execute your move.
So if you are going to be a dumbass . . .
Let’s review. A car moving at 60 mph in the HOV lane under dry conditions required 118 feet to come to a complete stop while skidding on the concrete roadway. So if you’re going to make an illegal jump into the HOV lane, you had better be a good judge of distance and know that the car already in the HOV lane is at least 118 feet behind you. But it’s worse than that, because it will take 1.6 seconds from the moment you decide to jump to the moment when your acceleration puts you in the HOV lane. In that time, a car in the HOV lane will go 136 feet. If you are then expecting that car to slam on its brakes and avoid hitting you, you have to plan even further ahead. Since the car in the HOV lane will travel 136 feet and then requires 118 feet to come to a complete stop, the minimum distance away that car should be when you decide to jump is 254 feet. Are you a good judge of distance to at least 254 feet? I’ll bet you’re not.
Oh, and guess what: the driver in the HOV lane is human, too. They also have a reaction time of at least 0.2 seconds, from the time they see you lane-jumping to the time they apply the brakes and go into a skid (unless they are anti-lock brakes, in which case the stopping distance will be elongated!). So that means you need to have judgement to a distance corresponding to 1.8 seconds (at least), which is 271 feet (your reaction time, plus their reaction time, plus the time it takes you to accelerate into the HOV lane, plus the time it takes for them to come to a complete stop by skidding on a dry concrete roadway).
Judging distance – objects in the rearview mirror . . .
Next time you check your mirror on the driver side, look for the message written on the mirror that reads, “Objects in the mirror may appear further away than they actually are.” That message tells you you’re really screwed, because the convex surface of the mirror causes a distortion of distance. This distortion allows you to see more than you otherwise would for a planar mirror, but it comes at a cost to your ability to judge distance.
Conclusions
Basic physics – energy conservation and forces – can teach us a lot about the quality of our decisions. From a legal perspective, it’s already a punishable action to jump into the HOV lane anywhere but at an entrance. From a physics perspective, you have to be able to be absolutely sure that any car behind you in the HOV lane is at least 271 feet away (and that’s the BARE MINIMUM to be safe – you probably want to be sure the car is more like 400-500 feet away to account for longer reaction times, lower acceleration of your vehicle, and the fact that the vehicle in the HOV lane is probably speeding (going faster than 60 mph). Are you a good judge of distance to 500 feet? Does the curving of the highway, combined with long lines of traffic behind you, even allow that kind of visibility?
People often argue that science cannot provide us with any kind of morality. I disagree. Jumping into the HOV lane, independent of human law, is a sign of complete immorality. By doing it, you are disregarding energy conservation and the effects this has on the lives of individuals in your car and in the cars in the HOV lane. Because energy must be conserved, and if it can’t all be put into working against friction, it will instead be put into compressing the structures of the cars involved in the subsequent collision.
For an independent view of this issue, see Ref. 3.
[1] http://en.wikipedia.org/wiki/Sliding_friction
[2] http://physics.info/acceleration/
[3] http://transportationblog.dallasnews.com/archives/2009/07/should-hovers-turn-in-cheaters.html
