Einstein’s Relativity

California State Standard:
Physics 1 h*: Students know Newton’s laws are not exact but provide very good approximations unless an object is moving close to the speed of light or is small enough that quantum effects are important.

Warm Up: {diagnostic} The speed of light is constant. What happens as an object’s velocity approaches the speed of light?

Key facts:

The speed of light in vacuum is known to be constant at 3.0×10^8m/s. The speed of light is actually slower in media such as glass but is never more than 3.0×10^8m/s.

The laws of physics are the same to all observers in inertial (non-accelerating) reference frames.

Relativity-Part-1
Who determined that the speed of light is a constant and that the luminiforous ether does not exist?

Einstein’s postulates result in a conflict. To resolve this conflict we need to understand that while the speed of light is constant, the flow of time varies with relative velocity

Advanced topic (optional)

Lesson 2
Warm Up: {from memory} Now that you’ve studied the video, revisit the question: What happens as an object’s velocity approaches the speed of light? (remember that the speed of light is a constant)
What is the numerical value of the speed of light?

Learning Objective: To use the Lorent-Lorenz coordinate transformation to calculate time dilation.

Vocabulary: Dilation –  noun – the action or condition of becoming or being made wider, larger, or more open

In the context of relativity, dilation refers to the stretching of time, ie clocks slow down.  You will also see the word “dilitation”. This is also correct.

The Lorent-Lorenz coordinate transformation:

This is frequently called the Lorenz transformation. Lorent and Lorenz developed this mathematical form. Later Einstein applied this transformation to the problem of special relativity.

Lorenz-transformation

Write down these transformations.

Notice that these transformations apply to the measurement of length (x’) as well as time (t’). We’ll deal with length in a subsequent lesson.

A simpler formula for time dilation is simply

t’ = γ t

where gamma  gamma

Consider an astronaut wearing a heart monitor. His heart beats normally at 60 beats per minute or 1 beat per second.

Calculate gamma and t’  assuming

1] v = 0.5 c

2] v = 0.707 c   {notice: 0.707 is (square root 2)/2}

3] v =  0.866c



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Rather than calculating time dilation for a few individual values of v, it’s often more useful to use a spreadsheet (Excel), build calculator and graph (chart) the data.
The Excel project is:
for v/c = 0, 0.1, 0.2,….0.9, calculate Gamma and t’. (recall t is the length of time it takes the astronaut’s heart to beat once – ie. 1 second, t’ is the dilated time observed by a stationary observer.

repeat the spreadsheet for v/c = 0.9, 0.99, 0.999, 0.9999, 0.99999, and 0.999999.
Next change the vertical axis (y value axis) to logarithmic, observe how the shape of the graph changes. It’s sometimes useful to plot data in this way (“semi-log” plot) see attached spreadsheet.

Relativistic Time Dilation

Time dilation and the muon.

Muon’s are produced in the upper atmosphere by collisions between high energy particles such as cosmic rays with heavier gas elements. Typically this happens at around 10km altitude as this practically speaking the top of the atmosphere (note: atmospheric pressure drops to 1/e by about 8 km, also known as the “e”-folding depth. So at 10 km, pressure is around 1/5th of sea level pressure).

Lesson 3:

Warm Up:  {FM} calculate how long at takes a particle to decend from 10km altitude to the earth (sea level) if it is moving at or near the speed of light.  Then calculate

Relativity-Part-2