4. Special Relativity

Vocabulary:

Photon – a particle of light.

The Twins Paradox:

The Twins paradox is a common thought experiment in special relativity. A modified version of this experiment is as follows:

The government wishes to conduct an experiment to prove, once and for all, whether special relativity is true or not. They recruit two twins, Chell and Shell. Chell enters a rocket ship and takes off into space at very fast speed; let it be 90% of the speed of light. Shell does nothing; she just lives normally. As per mission protocol, Chell flies out into space until she runs out of supplies. That is the end of her mission. She then creates a “portal.” This portal links with a portal that Chell made on earth before staring her mission, allowing her to magically return to where she started. After her 17 year mission, Chell steps out of her spaceship and is surprised to find that her sister, Shell, has aged 40 years!

Background:

The twin paradox relies on Einstein’s Theory of Special Relativity. Einstein supposed that the speed of light was constant, regardless of the reference frame of the observer. This result contradicts physical intuition. Using this assumption, he derived a method of relating physics across reference frames. In his 1905 paper, Einstein derived this significant result:

t is the elapsed time in the stationary coordinate system (Shell), τ is the elapsed time in the moving coordinate system (Chell), v is the velocity of the moving system (Chell), and c is the speed of light in a perfect vacuum. Apply this equation for Chell and Shell (i. e. substitute v = 0.9*c, and t = 40) and the twin paradox will appear.

Explanation:

An English translation of Einstein’s 1905 paper is available here. The purpose of this blog post is to present some of the results of the theory in a manner that can be easily understood, while at the same time retaining some faithfulness to the original physics.

A first step in this analysis will be to define a method of measuring time. All choices of units are arbitrary. For example, the English unit, foot, was literally the length of someone’s foot. Similarly, the units of time are arbitrary. For the sake of this analysis, time can then be measured as distance divided by time. For example, moving 10 miles at 10 miles per hour will be 1 hour.

Imagine a rod of length L with endpoints labeled A and B. The rod is oriented as shown in the diagram below; it moves towards the positive x direction with a constant speed v.

You, the observer, are moving next to the rod at with the exact same speed as the rod. Now imagine a photon being released from point A which is heading in the direction of point B. While the photon is moving towards B at speed c, point B is moving away from the photon at the speed v. The net velocity is difference of the two speeds, or c - v. Therefore, the time the photon needs to get from A to B, tAB, is defined:

Now let a photon move from B to A. The photon will move towards A with a speed c, but A is moving towards the photon at a speed v. The net speed will be c + v, and the time, tBA, is

The times are not the same! For all positive values of c and v, tAB will always be greater than tBA. As an illustration of this point, let c = 2 and v = 1, then tAB = L while tBA = 1/3*L. In everyday life, this difference is negligible. The speed of light is approximately 300 million m/s. The fastest commercial airliners move at speeds lower than Mach 1, or 340 m/s. For these values, tAB is only 0.00025 % larger than tBA, an amount so small that it is usually not worth mentioning. All results of special relativity can be derived in manner similar to, albeit more rigorous than, the above analysis.

Results:

We have definitively shown that time is asynchronous when dealing with moving reference frames. This is a central point of The Theory of Special Relativity, and one of the foundations for all of modern physics. Using this example as a starting point, one can begin to develop an intuition for relativistic physics.

References:
[1] A. Einsten, "On the Electodynamics of Moving Bodies."
[2] Wikipedia. Search "Special Relativity"
[3] Wikipedia. Search "Introduction to Special Relativity"

3. Rocket Science

Problem:

How do rockets work?

Background:

In popular culture, the phrases “rocket science” and “brain surgery” are used to represent any task that requires great intelligence or skill. The purpose of this post is to show that rocket science isn’t that hard. You can make rockets too! In addition to this, there are a few common misconceptions which I will clear up.

Solution:

As with most problems in physics, the first place to start is Newton’s laws. Newton’s 2^nd law is commonly stated as force equals mass times acceleration, F = ma. This statement is almost true. The true statement of Newton’s 2^nd is that the change in momentum is related to the force. Momentum is, in turn, the product of mass and velocity. A force can be generated by a change in velocity (acceleration) or a change in mass. The ingenious method employed by rockets is to use a mass change instead of an acceleration change to generate force. This can be illustrated by the example below.

A rocket uses its nozzle to accelerate the flow at exit to a high speed above the inlet velocity. Due to the shape of the nozzle the speed at exist must necessarily be higher than the speed at the inlet. (See solution section of this post for the proof.) Applying Newton’s 2^nd law to a fluid element and making simplifying assumptions relevant to this problem, we can derive this equation

F = mt*v +p*A (Equation 1)

Where mt, is the mass flow rate. It represents the rate at which the rocket is losing mass. The net momentum flux, mt*v, results in a force on the rocket (thrust) which accelerates the rocket forward. In addition, the pressure causes a force on the rocket which is proportional to the nozzle area. There is a common misconception that the pressure at the nozzle exit causes thrust. The equation demonstrates that the fluid flux, alone, is enough to produce force in the rocket.

Model Limitations:

A major limitation with the usefulness of Equation 1 is lack of information. It would be very difficult or impossible to know all of the values for velocity and pressure. However, given this information, the force can be calculated exactly.

2. How Airplanes Produce Lift

Problem:

How do airplanes produce lift?

Vocabulary:

Airfoil – The two dimensional image of an airplane’s wing. Imagine cutting a thin slice of the wing while looking from the side of the airplane.

Background:

The problem of how airplanes produce lift has been tackled many times. Most people know the basic theory. The shape of the foil causes air to move faster over the top of the wing than it moves over the bottom. Bernoulli’s Equation implies that pressure and speed are inversely related. Applying Bernoulli’s Equation, it can be seen that the pressure on upper portion of the wing must be lower than the pressure on the bottom of the wing. This pressure differential results in a net force that causes the airplane to move upwards. This explanation is correct at its core, but the model simplification leads to a few issues:

1) How can airplanes fly upside down? In the above model, the pressure differential will cause the airplane to plummet towards the earth.

2) How do airplanes or gliders with flat wings fly?

The answer is that the curvature is not the only factor that causes a pressure differential to occur over the wing. I hope to present an explanation that is rigorous in its physical derivation, but simple enough for the layman to understand.

Solution:

Before looking at flow over an airfoil, it is important to first look at the Law of Conservation of Matter. The Law of Conservation of Matter states that matter can be neither created nor destroyed, but it can change form. For a fluid like air, the law of conservation of matter can be expressed mathematically as:

d1 * v1 = d2 *v2

Where d represents the distance between the walls at a location, and v represents the speed at that location. The flow was assumed to be incompressible and one dimensional (namely the flow goes from right to left). In this context, incompressible means that you can’t “squeeze” a given volume of fluid down to a smaller size. This equation is called the continuity equation.

Now imagine a pipe of varying wall spacing, and apply the continuity equation. As the pipe wall spacing changes, the fluid speed must change as well. In the example below, setting d1=5, v1=3, and d2=3, we can derive v2=5.

Thus, the equation implies that as the wall spacing decreases, the fluid speed increases. This is a very significant result, and it is the basis of our solution.

Now the example can be extended to an airfoil. For generality, the airfoil will not be level: it will be at an angle of attack with respect to the freestream flow. This will help to demonstrate why flat wings can produce lift. The sketch below is an approximation of what streamlines would look like over the upper surface of an airfoil at an angle of attack.

Pretend that the regions labeled as a dotted lines are a walls. This is a valid assumption since there will be no flow into or out of these lines (they are horizontal streamlines). Now the continuity equation can be used. Focusing in on the top region, it can be seen that the flow must speed up while traveling over the top of the airfoil. The wall spacing is smaller over the top surface of the airfoil; therefore, the air must move faster. For the same reason, the flow must speed up as it goes along the bottom surface. Placing the airfoil at any angle such that air flow is faster over the top surface than over the bottom will produce lift.

Model Limitations:

This analysis assumed an idealized (1D) steady incompressible (irrotational) flow. In plainer words, it assumes that the airplane is flying below Mach 0.3 speed and that there are neither frictional nor “3D” effects. In general, none of these things are true in real aviation. While this invalidates the model in terms of rigorous mathematical application, I believe the model still serves as a useful tool to model trends for all types of flows. This is supported by the fact that airplanes fly.

1. Airplane on Treadmill

Problem:

Imagine an airplane on a treadmill. The treadmill’s speed is made to exactly match the lift-off speed of the airplane, and the treadmill is oriented such the airplane (moving at lift-off speed) would appear stationary with respect to the ground. Will the airplane be able to lift off?

Solution:

The airplane lifts off.

Background:

The problem that I am addressing today is well studied. Every few months someone posts the story on an internet forum or website, and people argue their perspectives on the issue. The problem grew to become so popular that it was featured on MythBusters [video]. The result is already known to many of you, but the reason why this result occurs is mostly unexplained. Most “explanations” of this problem stray too far from physics and instead rely on simplified intuitional models. I hope to present a solution which is rigorous in its physical derivation, but simple enough for the layman to understand.

The null hypothesis (that the airplane will not lift-off) is based on basic aerodynamic theory. If there is no airflow over the airplane’s wings then it cannot produce any lift. A stationary airplane will have no airflow, and, therefore, will not produce lift. All of this is true. The problem with the null hypothesis is the assumption that the airplane will remain stationary. Using basic physics, I will demonstrate that the airplane must necessarily accelerate forward.

Proof:

Before beginning the proof, a basic understanding of Newton’s laws of motion is required.

Newton’s 2^nd Law is that when an object is acted upon by a net force, the object will move with an acceleration which is related to the mass of the object and the magnitude of the force.

Newton’s 3^rd Law is that every force causes an equal and opposite force to occur. A good example of this is jumping. You push against the ground with your legs; the ground pushes you back which causes you to move upwards.

It would be illustrative to first talk about a car moving on a treadmill like the one mentioned in this problem. We will consider the system (the object that we are solving the equations of physics for) to be the car and the treadmill together. This is counterintuitive (the natural assumption is to make the car be the object), but it is an important and useful assumption for this problem. Let’s take a look at the forces involved in the problem. For the sake of this analysis, we need only look at forces in the horizontal direction.

The car moves by turning its wheels. The motion of the wheels creates a force on the treadmill (friction). By Newton’s 3^rd, the treadmill creates an opposing force on the car. The forces acting on the car and the treadmill are equal and opposite, so they will always cancel each other out. There is no net force. By Newton’s 2^nd, if there is no net force then there is no acceleration. The car remains stationary.

Now, solve the same system for the airplane. The airplane/treadmill is a more complex system than the car/treadmill because it has thrust in addition to frictional forces.

Just like the car, the friction generated by the airplane will cancel with the friction generated by the treadmill. The difference is the existence of the extra force, thrust. Apply Newton’s 2^nd. There is a net force (the thrust) and therefore there will be some net acceleration. The airplane will move! This video by neodocneodoc demonstrates my point.

Model Limitations:

There are practical limits to the amount of thrust an airplane’s engine can output. The simplified model above assumes that the airplane’s engine can produce enough thrust to accelerate to the speed needed by the airplane to lift off from the treadmill. In reality, there is a certain upper limit speed that the airplane can reach which depends on its engine type. This upper limit speed may not be enough to take off from a very fast moving treadmill.

Aerodynamic drag was ignored in this analysis because it is usually small compared to the other forces. This is a legitimate assumption for most situations, but if you drive a supersonic car then you will need to include drag effects. (^_^)

Next time: How airplanes produce lift.

Why make a blog?

The purpose of this blog is to create a written record of my progress as a science writer. The blog will attempt to use as few words as possible to precisely describe phenomenon from the fields of science, technology, and mathematics. The use of  the word "precisely" is significant. There are many people who write about science, but there are few who write with precision. Ofttimes, critical information is left out of the physical model. This simplifies explanation, but causes confusion when simple approximations don't live up to real world situations. For this blog, all relevant theorems, laws, and assumptions will be addressed in each article. This blog will also strive to be mathematically friendly; knowledge of basic high school algebra should be sufficient for comprehension.