### SAT Physics Relativity - Special Theory Of Relativity

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**SAT Physics Relativity - Special Theory Of Relativity**

**SPECIAL THEORY OF RELATIVITY**

Until Einstein’s paper in 1905, most physicists believed that the universe was filled with an invisible medium called ether. They reasoned that ether was necessary in order for light, considered by most physicists to be a wave, to propagate through space. However, experiments such as the famous Michelson-Morley experiment all failed to prove the existence of ether.

Einstein viewed the problem in a completely different manner and proposed an explanation for these failed experiments. He viewed light as a quantum particle (later named a photon) that traveled with a specific speed in a vacuum, c = 3 × 10

^{8}meters per second. Einstein then suggested that the speed of light is the same for all

**inertial reference frames.**An inertial reference frame is a frame of reference moving at a constant velocity. (Inertia is the tendency of objects to continue moving at constant velocity.) Einstein stated that the speed of light and all laws of physics are the same for any inertial reference frame. This is known as Einstein’s

**first postulate of special relativity.**

In addition to the speed of light being the same for all inertial reference frames, Einstein stated that even if light were emitted from a moving source, it would continue to have a velocity of c. This is known as Einstein’s

**second postulate of special relativity**. The second postulate is counterintuitive to the laws of motion that describe the relative motion of two moving objects. Consider a ball being thrown forward from a moving vehicle. The true speed of the ball is the speed of the throw plus the speed of the moving vehicle. This is not so with light. Light beams from a car’s headlight travel at c regardless of the speed of the car. The true speed of light is always c when measured in any inertial reference frame.

Einstein’s special relativity is a special case since inertial reference frames are not accelerating. In 1915, Einstein proposed a theory of general relativity that encompassed accelerating reference frames in addition to inertial reference frames. The general theory of relativity, however, is beyond the scope of the SAT Subject Test in Physics.

**TIME, LENGTH, AND MASS**

Using the assumption that light travels at a constant velocity of c for any inertial reference frame, several noticeable effects will happen to time, length, and mass at speeds approaching c. Note that the formulas in the following sections are provided for context. You will probably not have to solve them on the actual examination as doing so would require a calculator. The exam will instead ask generalized questions regarding the effects on time, length, and mass when objects are moving relative to an observer on the object or a stationary observer watching the object speed by.

**Time Dilation**

Picture a person watching a moving object. An observer will usually interpret his or her own motion as being stationary and will perceive the other object as moving. Both the observer (stationary) and the object (moving at constant velocity) are equipped with clocks. The observer will see his or her own clock as running normally but will see the clock on the moving object as running slowly. This effect is known as

**time dilation**. The amount that the clock appears to be running slowly can be calculated with the following equation:

Time, t

_{0},is the time on the clock of the stationary observer, which appears to be running normally. However, when the stationary observer looks at the clock on the moving object, it will have a different, slower time, t. The velocity of the moving object is v, and c is the speed of light. If the moving object has a small velocity, both clocks will appear to be running at the same time, t= f

_{0}. The time difference between the two clocks will be negligible and the effects of time dilation will go unnoticed.

The speeds of manmade objects are too slow for dilation effects to appear. As a result, we are unaware that dilation actually occurs. However, as the speed of the object nears the speed of light, the effects of time dilation increase dramatically. When an object moves at 99.5% of the speed of light (v = 0.995c), the time difference in the clocks is substantial.

To an outside stationary observer, events in the moving frame (time t) appear to take 10 times as long as the same events in the observer’s stationary frame (time t

_{0}). The events in the moving frame appear to be running at one-tenth the speed of events in the observer's frame of reference. The clock and a person on the moving object would appear to be moving in slow motion. At the speed of light, v = c, the clock on the moving object would appear to stop entirely.

Since the incredible speeds needed to observe time dilation are unlikely to be attained by humans, evidence of time dilation has been found by examining tiny, high-speed particles. Muons are negatively charged elementary particles with a lifespan of about 2.2 × 10

^{-6 }seconds. They are formed by the collision of cosmic rays with the atmosphere. Even though they travel at nearly the speed of light (0.99c), the distance from the atmosphere to Earth is such that they should decay before they hit the ground. Yet they are detected at the surface of Earth. This indicates that muons are experiencing time dilation.

**Length Contraction**

When an object moves, the length of the object and anything moving with the object appears to contract in length. The observed length is calculated with the following equation:

You should note that the actual length affected by motion is the length that matches the direction of motion. If an object is moving to the right, only length along the x-axis is affected. The height of the object in the y-direction and its depth in the z-direction remain unchanged. In the formula above, length is the object’s rest length. This is the length of the object measured when the object is at rest or by someone moving at the same speed as the object. Length L is the length measured by a stationary observer, someone not moving with the object. At low velocities, v, these lengths are nearly identical, L ≈ L

_{0},and the length contraction is not noticeable.

_{0}. At the speed of light, v = c, length would become 0, destroying three-dimensional space.

**Mass Effects**

Mass, like time and length, is also affected by the motion of objects. As the speed of an object approaches c, the mass of the object, m, increases.

Mass m is the increased mass that a stationary observer would perceive. Mass m

_{0}is the object’s rest mass. The rest mass is the mass of the object when it is stationary. It is also the mass measured by an observer traveling at the same speed as the object. As with time and length, the effects are unnoticed at the low speeds that dominate human experience, m ≈ m

_{0}.

As velocity, v, approaches c, the mass of an object will increase. At the speed of light, v = c, the mass of the object will become infinite. According to Newton’s second law (F = ma), an infinite force would be required to accelerate the object. Therefore, the speed of light, c, is the speed limit for material objects. No material object (made of matter) can travel faster than light.

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