krit.club logo

Space, Time and Motion - Galilean and Special Relativity

Grade 11IBPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

An inertial frame of reference is a frame in which Newton's First Law of motion holds (the frame is not accelerating).

•

Galilean Relativity: In classical mechanics, velocities are additive. If an object moves at velocity uu in a frame SS, its velocity u′u' in frame S′S' moving at velocity vv relative to SS is given by u′=u−vu' = u - v.

•

Postulates of Special Relativity: 1. The laws of physics are the same in all inertial frames of reference. 2. The speed of light in a vacuum, cc, is constant for all observers, regardless of the motion of the source or the observer.

•

The Lorentz Factor: Denoted by γ\gamma, it measures the degree of relativistic effects. It is defined as γ=11−v2c2\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}. As vv approaches cc, γ\gamma approaches infinity.

•

Time Dilation: The time interval between two events occurring at the same place in their rest frame is called proper time Δt0\Delta t_0. An observer moving relative to that frame will measure a longer time interval Δt=γΔt0\Delta t = \gamma \Delta t_0.

•

Length Contraction: The length of an object measured by an observer at rest relative to the object is the proper length L0L_0. An observer moving parallel to the object's length at velocity vv will measure a shorter length L=L0γL = \frac{L_0}{\gamma}.

•

Simultaneity: Two events that are simultaneous in one frame of reference are not necessarily simultaneous in another frame moving relative to the first.

📐Formulae

u′=u−vu' = u - v

γ=11−v2c2\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}

Δt=γΔt0\Delta t = \gamma \Delta t_0

L=L0γL = \frac{L_0}{\gamma}

x′=γ(x−vt)x' = \gamma(x - vt)

t′=γ(t−vxc2)t' = \gamma(t - \frac{vx}{c^2})

💡Examples

Problem 1:

A muon is traveling at v=0.99cv = 0.99c relative to the laboratory. Its mean lifetime in its own rest frame is Δt0=2.2×10−6 s\Delta t_0 = 2.2 \times 10^{-6} \text{ s}. Calculate the lifetime of the muon as measured by a scientist in the laboratory.

Solution:

  1. Calculate γ\gamma: γ=11−0.992≈7.09\gamma = \frac{1}{\sqrt{1 - 0.99^2}} \approx 7.09.
  2. Use the time dilation formula: Δt=γΔt0=7.09×2.2×10−6 s≈1.56×10−5 s\Delta t = \gamma \Delta t_0 = 7.09 \times 2.2 \times 10^{-6} \text{ s} \approx 1.56 \times 10^{-5} \text{ s}.

Explanation:

Because the muon is moving at a significant fraction of the speed of light relative to the lab, its internal 'clock' appears to run slow to the laboratory observer, resulting in a longer measured lifetime.

Problem 2:

A meter stick (L0=1.00 mL_0 = 1.00 \text{ m}) flies past an observer at a speed of v=0.80cv = 0.80c. What is the length of the stick as measured by the observer?

Solution:

  1. Calculate γ\gamma: γ=11−0.802=11−0.64=10.36=10.6=1.67\gamma = \frac{1}{\sqrt{1 - 0.80^2}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6} = 1.67.
  2. Apply length contraction: L=L0γ=1.001.67≈0.60 mL = \frac{L_0}{\gamma} = \frac{1.00}{1.67} \approx 0.60 \text{ m}.

Explanation:

Length contraction only occurs in the direction of motion. To the stationary observer, the meter stick appears shortened to 60 cm60 \text{ cm}.

Galilean and Special Relativity - Revision Notes & Key Formulas | IB Grade 11 Physics