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Ch. 36 - The Special Theory of Relativity
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 36 - The Special Theory of RelativityProblema 92
Capitolo 35, Problema 92

Using Example 36–2 as a guide, show that for objects that move slowly in comparison to c, the length contraction formula is roughly ℓ ≈ ℓ₀ (1 - 1/2 v²/c²) . Use this approximation to find the “length shortening” ∆ℓ = ℓ₀ - ℓ of the train in Example 36–6 if the train travels at 100 km/h (rather than 0.92c).

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Identify the given values and relevant formulas. In this case, the original length of the train (ℓ₀) and its speed (v) are given. The speed of light (c) is a constant. The length contraction formula in special relativity is ℓ = ℓ₀√(1 - v²/c²).
Since the train moves much slower than the speed of light, we can use the binomial approximation for the square root in the length contraction formula when v is much smaller than c. This approximation is √(1 - x) ≈ 1 - 1/2 x for small x. Here, x corresponds to v²/c².
Substitute v²/c² into the binomial approximation to modify the length contraction formula: ℓ ≈ ℓ₀(1 - 1/2 v²/c²).
Calculate the approximate contracted length ℓ using the modified formula by substituting the values of v and c into the formula. Remember to convert the speed of the train from km/h to m/s for consistency with the units of c (which is in m/s).
Finally, find the length shortening ∆ℓ by subtracting the contracted length ℓ from the original length ℓ₀: ∆ℓ = ℓ₀ - ℓ.

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Length Contraction

Length contraction is a phenomenon in special relativity where the length of an object moving at a significant fraction of the speed of light (c) is measured to be shorter than its proper length (ℓ₀) when at rest. The formula for length contraction is ℓ = ℓ₀√(1 - v²/c²), where ℓ is the contracted length, v is the object's velocity, and c is the speed of light. This effect becomes noticeable as the object's speed approaches c.
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Length Contraction

Approximation for Low Speeds

For objects moving at speeds much lower than the speed of light, the effects of relativity can be approximated using a simplified formula. Specifically, when v is much less than c, the length contraction can be approximated as ℓ ≈ ℓ₀(1 - 1/2 v²/c²). This approximation allows for easier calculations without needing to use the full relativistic equations, making it practical for everyday speeds.
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Calculating Length Shortening

To find the length shortening (∆ℓ) of an object, one can use the difference between its proper length (ℓ₀) and its contracted length (ℓ). The formula is ∆ℓ = ℓ₀ - ℓ. By substituting the approximate length contraction formula into this equation, one can calculate how much shorter the object appears when moving at a given speed, such as 100 km/h in the context of the train example.
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