(455) HL Length contraction [IB Physics HL]

Length Contraction

• Length contraction: occurs in fast-moving objects; they appear shorter relative to an observer at rest.

• Example: When moving in a spaceship, you won't notice your ship's contraction, but observers on Earth will.

  • Proper Length: The length of an object as measured when at rest, denoted as ( L_0 ).

  • Observed Length: The length measured by someone for whom the object is not at rest, denoted as ( L ).

  • Length contraction formula: ( L = \frac{L_0}{\gamma} ) where ( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} )

Example Calculation

• While traveling at ( 0.8c ):

  • Proper Length of spaceship: ( L_0 = 18 \text{ m} )

  • Calculate ( \gamma ):

    • ( \gamma = \frac{1}{\sqrt{1 - (0.8)^2}} = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6} \approx 1.67 )

  • Observed Length: ( L = \frac{18 m}{1.67} \approx 10.8 m )

    • To Earth observers, the spaceship appears contracted to ( 10.8 m ).

Magical Barn Example

• A spaceship is ( 2 m ) long and a magical barn is ( 1.6 m ).

• Question: What speed is needed for the spaceship to fit in the barn with both doors closed?

• Use length contraction formula:

  • ( L = \frac{L_0}{\gamma} ) where ( L = 1.6 m ) (for barn), ( L_0 = 2 m ) (for spaceship).

• Solve for ( \gamma ):

  • ( 1.6 = \frac{2}{\gamma} ) gives ( \gamma = \frac{2}{1.6} = 1.25 )

• Set ( \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} ):

  • Squaring and rearranging yields ( \frac{v^2}{c^2} = 1 - (\frac{1}{1.25})^2).

  • Find ( V ):

    • Result: ( V \approx 0.6c )

• Conclusion: At ( 0.6c ), the ( 2 m ) spaceship will appear ( 1.6 m ) and can fit in the barn.