Worksheet: Length Contraction

In this worksheet, we will practice calculating the change in the observed length as measured by an observer moving with a relative velocity.


A spaceship that has a proper length of 2.00Γ—10 m moves at 0.970𝑐 relative to Earth. What is the spaceship’s length as measured by an observer on Earth?


An observer located at the origin of an inertial frame 𝑆 sees the flash of a flashbulb occur at the position π‘₯=150km and at the time 𝑑=4.50Γ—10οŠͺs. The system π‘†οŽ˜ is moving along the π‘₯-direction of 𝑆 at a velocity of 0.600𝑐.

At what time in the π‘†οŽ˜ system did the flash occur?

  • A2.13Γ—10οŠͺ s
  • B2.64Γ—10οŠͺ s
  • C3.37Γ—10οŠͺ s
  • D3.91Γ—10οŠͺ s
  • E1.88Γ—10οŠͺ s

At what π‘₯-position did the flash occur as measured in the π‘†οŽ˜ system?


A spaceship (A) is moving at speed 𝑐/2 with respect to another spaceship (B). Observers in A and B set their clocks so that the event at (π‘₯,𝑦,𝑧,𝑑) of turning on a laser in spaceship B has coordinates (0, 0, 0, 0) in A and also (0, 0, 0, 0) in B. A photon from the laser in spaceship B arrives at (π‘₯=1.00,0,0)m at 𝑑=0 in the frame of ship A.

What will be the time value 𝑑′ at which the observer in spaceship B measures the photon’s arrival?

  • A2.15Γ—10 s
  • B2.08Γ—10 s
  • C2.29Γ—10 s
  • D1.93Γ—10 s
  • E2.40Γ—10 s

What displacement value π‘₯β€² will the observer in spaceship B measure for the position of the arriving photon?


An astronaut measures the length of his spaceship to be 100 m, while an observer on Earth measures it to be 25.0 m.

Find the Lorentz factor 𝛾 that relates the values of measurements by the astronaut and the observer on Earth.

What is the velocity of the spaceship relative to earth?

  • A0.968𝑐
  • B0.922𝑐
  • C0.984𝑐
  • D0.867𝑐
  • E0.900𝑐


How fast should an athlete run for them to perceive a 200 m race as a 200 yd race given that 1 yd = 0.9144 m?

  • A0.287𝑐
  • B0.293𝑐
  • C0.395𝑐
  • D0.164𝑐
  • E0.405𝑐


Spaceship A is moving at speed 𝑐3 with respect to another spaceship B. A rod of length 2 m is laid out on the π‘₯-axis in the reference frame of B from the origin to (2.00, 0.00, 0.00). What is the length of the rod as measured by an observer in the reference frame of spaceship A? Give your answer to 3 significant figures.


How fast would a 5.000 m long sports car have to be going past an observer for the observer to measure its length as 4.500 m?

  • A0.5217𝑐
  • B0.3160𝑐
  • C0.1898𝑐
  • D0.2537𝑐
  • E0.4359𝑐

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