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).

Verified step by step guidance

1. 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).
2. 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.
3. Substitute v²/c² into the binomial approximation to modify the length contraction formula: ℓ ≈ ℓ₀(1 - 1/2 v²/c²).
4. 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).
5. Finally, find the length shortening ∆ℓ by subtracting the contracted length ℓ from the original length ℓ₀: ∆ℓ = ℓ₀ - ℓ.

Key Concepts

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.

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.

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.