PHYS 221 Recitation

Kevin RalphsWeek 12

Overview

• HW Questions• Chapter 27: Relativity– History of Special Relativity (SR)– Postulates of SR– Time Dilation– Simultaneity– Length Contraction– Velocity Addition– Mass, Energy and Momentum

History of SR

• Historically, the nature of light was a source of much debate– Did it travel instantaneously?– Is it a wave or a particle?

• If a wave, how do we explain reflection and aberration of starlight?

• If a particle, how do we explain refraction?

• Humans tend to explain new experiences through old ones and multiple theories and interpretations were given centering around the idea of an aether that mediated light’s transmission

History of SR

• In trying to understand this aether, most of the results of SR and its mathematics were already developed prior to Einstein’s proposal

• Einstein’s contribution was to explain all these results from two postulates with no reference to an aether

Postulates of SR

1. All the laws of physics are the same in all inertial frames

2. The speed of light in a vacuum is a constant, independent of the motion of the light source and all observers

An inertial frame is one that is not undergoing any kind of acceleration

Time Dilation

• What does it tell me?– A moving clock appears to run slower

: The perceived elapsed time: The elapsed time in the rest frame of the

clock; aka proper time

Time Dilation

• Why should I care?– The -factor is normally very close to unity, but as

velocities increase it can become significant– So any observation of a fast moving clock would

need to take this into account: for example, GPS satellites or decay of quickly moving particles

Simultaneity

• In general, observers in different inertial frames will not agree on the order of events

• However, all observers will agree on the order of events that occur at the same position or are causally connected

Length Contraction

• What does it tell me?– A moving meterstick appears to be shorter

: The perceived length: The length in the rest frame of the

meterstick; aka the proper length

Velocity Addition

• We can always keep increasing the velocity of an object, but there’s a limit to how fast an object goes

• Obviously, the way we normally add velocities has no such limit

Velocity Addition

• The correct way to add velocities is as follows:

• The formula may seem a little arbitrary, but it all comes from the geometry of a hyperbola which governs all the mathematics of SR

Relativistic Momentum

• We normally think of momentum as

• Velocities cannot exceed that of light in a vacuum so our formula would imply that momentum has a limit in SR

• We don’t measure any such limit• The correct way to define momentum is:

Mass and Energy

• Inertial mass is usually understood as the constant of proportionality that tells us the relationship between the forces on a body and its acceleration (i.e. F = ma)

• Unfortunately, due to SR this definition becomes frame dependent

• So in SR it becomes better to think of mass as a constant of proportionality showing the relationship between its intrinsic or rest energy and the square of the speed of light

rest energy

Mass and Energy

• The total energy of an object then becomesTE =

and the kinetic energy isKE =

Quiz QuestionConsider the motion of a muon as it moves through he Earth’s atmosphere. A particular muon is created 15,000 m above the Earth’s surface and just reaches the ground before it decays (2.197e-6 s). What is the speed of the muon?

a) 6.818e9 m/sb) 2.995e8 m/sc) 2.997e8 m/sd) 6.792e9 m/se) 2.986e8 m/s

Quiz QuestionIn studies of elementary particles such as electrons, physicists use machines called “accelerators” in which the particles move at very high speeds. Suppose an electron in one of these machines has a speed of 0.9999c and it travels along a straight line for a distance of 500 m as measured by physicists doing an experiment. How long is the accelerator as measured by a person riding along with the electron?

a) 22.3 mb) 11.1 kmc) 35.3 kmd) 7.07 m

Quiz Question

At what value of is the relativistic momentum of a particle twice its classical (Newton’s law) momentum?

a) 2.60e9 m/sb) 1.50e8 m/sc) 2.78e8 m/sd) 2.60e8 m/se) 3.10e8 m/s

Quiz Question

In 2005, the total energy “consumed” by people on Earth was about J. About 85% of this was from burning of fossil fuels. Equating his energy gained by burning fossil fuel in 2005 with mass (according to the mass-energy relation of relativity), how much mass was converted to energy?

a) 4.72e3 kgb) 1.41e12 kgc) 1.67e12 kgd) 5.55e3 kg

Quiz Questions

An asteroid of mass 2500 kg has a (relativistic) kinetic energy of J. What is the speed of the asteroid?

a) 2.6e8 m/sb) 2.9e8 m/sc) 2.4e8 m/sd) 3.5e8 m/s