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 . 0 0 Γ— 1 0 2 m, moves at 0 . 9 7 0 𝑐 relative to the Earth. What is the spaceship’s length as measured by an observer on the Earth?

Q2:

An observer located at the origin of an inertial frame 𝑆 sees the flash of a flashbulb occur at the position π‘₯ = 1 5 0 k m and at the time 𝑑 = 4 . 5 0 Γ— 1 0  οŠͺ s . The system 𝑆  is moving along the π‘₯ -direction of 𝑆 at a velocity of 0 . 6 0 0 𝑐 .

At what time in the 𝑆  system did the flash occur?

  • A 2 . 6 4 Γ— 1 0  οŠͺ s
  • B 2 . 1 3 Γ— 1 0  οŠͺ s
  • C 3 . 3 7 Γ— 1 0  οŠͺ s
  • D 1 . 8 8 Γ— 1 0  οŠͺ s
  • E 3 . 9 1 Γ— 1 0  οŠͺ s

At what π‘₯ -position did the flash occur as measured in the 𝑆  system?

  • A βˆ’ 1 0 1 km
  • B βˆ’ 4 8 . 9 km
  • C 119 km
  • D 178 km
  • E βˆ’ 8 2 . 6 km

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 . 0 0 , 0 , 0 ) m at 𝑑 = 0 in the frame of ship A.

What will be the time value 𝑑 β€² at which the observer in spaceship B measures the photon’s arrival?

  • A 2 . 1 5 Γ— 1 0 βˆ’ 9 s
  • B 2 . 0 8 Γ— 1 0 βˆ’ 9 s
  • C 2 . 2 9 Γ— 1 0 βˆ’ 9 s
  • D 1 . 9 3 Γ— 1 0 βˆ’ 9 s
  • E 2 . 4 0 Γ— 1 0 βˆ’ 9 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?

  • A 0 . 9 6 8 𝑐
  • B 0 . 9 2 2 𝑐
  • C 0 . 9 0 0 𝑐
  • D 0 . 8 6 7 𝑐
  • E 0 . 9 8 4 𝑐

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?

  • A 0 . 1 6 4 𝑐
  • B 0 . 2 9 3 𝑐
  • C 0 . 2 8 7 𝑐
  • D 0 . 4 0 5 𝑐
  • E 0 . 3 9 5 𝑐

Q6:

Spaceship A is moving at speed 𝑐 3 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?

  • A 0 . 3 1 6 0 𝑐
  • B 0 . 1 8 9 8 𝑐
  • C 0 . 5 2 1 7 𝑐
  • D 0 . 4 3 5 9 𝑐
  • E 0 . 2 5 3 7 𝑐

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