DP IB Physics: SL
A. Space, time and motion
A.5 Galilean and special relativity
DP IB Physics: SLA. Space, time and motionA.5 Galilean and special relativityLinking questions: |
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| a) | How are equations of linear motion adapted in relativistic contexts? |
| b) | Why is the equation for the Doppler effect for light so different from that for sound? |
| c) | Special relativity places a limit on the speed of light. What other limits exist in physics? (NOS) |
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a) How are equations of linear motion adapted in relativistic contexts?
- Solution:
- Equations of linear motion are modified in relativistic situations to take special and general relativity into consideration.
- These adjustments entail taking into account the curvature of spacetime, particularly in general relativity, and altering the equations to incorporate the relativistic component, which is dependent on the object’s velocity in relation to the speed of light.
- ⇒ Lorentz factor γ:
- The Lorentz factor is the fundamental modifier in relativity:
- [math]\gamma = \frac{1}{\sqrt{1 – \frac{v^2}{c^2}}}[/math]
- The majority of relativistic equations include this component. As v → c, γ → ∞
- ⇒ Adaptations of equations:
| Classical Mechanics | Relativistic version |
|---|---|
| Momentum: [math]p = mv[/math] | [math]p = γmv[/math] |
| Kinetic energy [math]KE = \frac{1}{2}mv^2[/math] | [math]KE = (γ – 1)mc^2[/math] |
| Total Energy: [math]E = KE[/math] | [math]E = γmc^2[/math] |
| Force:[math]F = ma[/math] | [math]F = \frac{d}{dt}(\gamma m v)[/math] |
| Energy-Mass Equivalence | [math]E = mc^2 [/math] (rest energy) |
- The concept of constant mass is employed in relativity to manage motion-related phenomena, and mass is not constant in terms of how it impacts momentum and energy.
- Four-momentum:
- The four-momentum is a conserved four-vector that is created in special relativity by combining momentum and energy.
- Energy that is relativistic:
- The equation links a particle’s mass and momentum to its total energy.
- [math]E^2 = (pc)^2 + (mc^2)^2[/math]
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b) Why is the equation for the Doppler effect for light so different from that for sound?
- Solution:
- The main reason why the Doppler effect equations for light and sound differ is because light, being an electromagnetic wave, does not require a medium to pass through.
- Because of this variation in wave propagation, different factors influence the perceived frequency shift in each instance.
- The relative velocities of the source, the observer, and the medium (such as air) all matter when it comes to sound. Only the relative velocity between the source and the observer is taken into account for light.
- Sound waves rely on a medium like air or water to propagate, and their speed is regulated by the medium’s qualities.
- In contrast, light waves can move in a vacuum, and their speed is constant regardless of the medium, according to the theory of special relativity.
- Relative Motion:
- While the Doppler effect for sound depends on the relative motion of the source, observer, and medium, the Doppler effect for light solely depends on the relative velocity between the source and observer.
- Special Relativity:
- Comprehension the Doppler effect for light requires a comprehension of the special theory of relativity.
- It requires that all observers travel at the same speed of light, independent of their relative motion, which has an impact on the computation of the Doppler shift.
- Figure 1 Special theory of relativistic
- ⇒ Doppler effect for sound:
- As mechanical waves, sound waves require a medium to travel through, such as air, water, or solid objects. The perceived frequency varies depending on how the source or observer moves in relation to the medium:
- [math]f’ = f \left( \frac{v \pm v_0}{v \pm v_s} \right)[/math]
- – [math]v_s = [/math]Speed of the source (Positive if moving toward the observer)
- – [math]v_o = [/math]Speed of the observer (Positive if moving toward the source)
- – [math]v =[/math] Speed of sound in the medium
- – [math]f^’ =[/math] Observed frequency
- – [math]f = [/math]Original frequency
c) Special relativity places a limit on the speed of light. What other limits exist in physics? (NOS)
- Solution:
- Our perception of the universe is shaped by numerous fundamental boundaries. These boundaries represent fundamental ideas about how nature functions and are not merely pragmatic or technological.
- There are other limits in physics outside the special relativity-established speed of light limit. These consist of:
- The Planck scales:
- This scale shows the maximum amount of time and space that can be compressed. The laws of special relativity may disintegrate at this scale, according to theories of quantum gravity.
- Quantum limits:
- According to quantum mechanics, there are restrictions on our understanding of physical systems.
- Heisenberg Uncertainty Principle, for example, states that the more precisely you know the position of a particle, the less precisely you know its momentum.
- Absolute zero-Temperature limit:
- Particles move very little at the lowest temperature of 0 Kelvin (–273.15°C).
- The Third Law of Thermodynamics states that it would take infinite steps or energy to achieve absolute zero, making it physically impossible.
- Heisenberg’s Uncertainty Principe:
- The age of the cosmos and the speed of light set limits on the observable universe.
- Beyond the cosmic horizon (around 46 billion light-years), we are unable to observe or communicate with anything.
- In cosmology, this is an observational boundary.
- Planck Scale – Limits of current theories:
- At high energies and very small scales (~ [math]10^{-35}[/math] m), physics moves into the Planck regime, where quantum gravity effects predominate.
- Existing theories such as quantum mechanics and general relativity fail.
- This highlights the shortcomings of existing scientific models and motivates the pursuit of a cohesive theory.