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.

Q1:

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?

Q2:

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?

Q3:

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?

Q4:

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

Q5:

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

Q6:

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.

Q7:

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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