Worksheet: Length Contraction

In this worksheet, we will practice calculating the change in the length of an object that is moving at a high speed relative to an observer.

Q1:

The Minkowski diagram shows two inertial reference frames: frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed of 0 . 2 𝑐 relative to 𝑆 . The two red lines are the world lines for the ends of a metal rod that is stationary in 𝑆 . The green lines show lines of simultaneity in each reference frame. Five events are shown on the diagram. For which two events would taking the difference in their 𝑥 coordinates give the length of the rod in 𝑆 ?

  • A 𝑃 and 𝑅
  • B 𝑄 and 𝑅
  • C 𝑃 and 𝑄
  • D 𝑅 and 𝑇
  • E 𝑊 and 𝑇

Q2:

A spaceship that is 300 m long in its rest frame is moving away from Earth at half the speed of light. Does the spaceship appear longer than 300 m, shorter than 300 m, or exactly 300 m long in the rest frame of Earth?

  • AShorter than 300 m long
  • BLonger than 300 m long
  • CExactly 300 m long

Q3:

The given Minkowski diagram shows three inertial reference frames: frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 , frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 , and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed of 0 . 3 𝑐 relative to 𝑆 and frame 𝑆 moves at a speed of 0 . 6 𝑐 relative to 𝑆 . The two red lines are the world lines for the ends of a metal rod that is stationary in 𝑆 . The green lines show lines of simultaneity in each reference frame. Six events are shown in the diagram. For which two events would taking the difference in their 𝑥 -coordinates give the length of the rod in 𝑆 ?

  • A 𝑃 and 𝑇
  • B 𝑊 and 𝑌
  • C 𝑃 and 𝑄
  • D 𝑄 and 𝑇
  • E 𝑃 and 𝑅

Q4:

A muon moves at a speed of 0 . 9 9 𝑐 along a linear particle accelerator. In the rest frame of the muon, it travels 650 m along the accelerator before decaying. How far did the muon travel along the accelerator in the rest frame of the accelerator? Give your answer, in kilometers, to 3 significant figures.

Q5:

A spaceship moves from Earth to the Alpha Centauri system. The spaceship moves at a constant speed of 0 . 9 𝑐 away from Earth. In the rest frame of Earth, the Alpha Centauri system is 4.37 light years away. In the rest frame of the spaceship, how far is this distance measured to be? Give your answer in light years to 3 significant figures.

Q6:

The given Minkowski diagram shows two inertial reference frames: frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed of 0 . 4 𝑐 relative to 𝑆 . The two red lines are the world lines for the ends of a metal rod that is stationary in 𝑆 . The length of the rod in 𝑆 , Δ 𝑥 , is 2 m and is equal to the difference in the 𝑥 -coordinate of events 𝑃 and 𝑅 . What is the difference in the 𝑥 -coordinate between events 𝑃 and 𝑄 in frame 𝑆 ? Give your answer to 3 significant figures.

Q7:

At what speed would a rigid metal bar have to move past you in order to have an apparent length 0.6 times its length in its rest frame? The bar is aligned parallel to its velocity. Give your answer as a multiple of the speed of light, 𝑐 .

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

Q8:

Sirius A, a star, is 8.66 light years away from Earth in the rest frame of Earth. At what speed relative to Earth would a spaceship have to travel for this distance to appear to be just 0.5 light years away? Give your answer as a multiple of the speed of light, 𝑐 , to 3 significant figures.

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

Q9:

A charged particle moves at a speed of 0 . 6 𝑐 along a linear accelerator. In the rest frame of the accelerator, the accelerator is 2 km long. How long is the accelerator in the rest frame from the charged particle?

Q10:

A spaceship that is 200 m long in its rest frame is moving toward Earth at half the speed of light. Does the spaceship appear longer than 200 m, shorter than 200 m, or exactly 200 m long in the rest frame of Earth?

  • ALonger than 200 m long
  • BExactly 200 m long
  • CShorter than 200 m long

Q11:

The given Minkowski diagram shows two inertial reference frames: frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed of 0 . 4 𝑐 relative to 𝑆 . The two red lines are the world lines for the ends of a metal rod that is stationary in 𝑆 . The length of the rod in 𝑆 , Δ 𝑥 , is 8 m and is equal to the difference in the 𝑥 -coordinate of events 𝑃 and 𝑄 . What is the difference in the 𝑥 -coordinate between events 𝑃 and 𝑅 in frame 𝑆 ? Give your answer to 3 significant figures.

Q12:

The given Minkowski diagram shows two inertial reference frames: frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with the axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed of 0 . 2 𝑐 relative to 𝑆 . The two red lines are the world lines for the ends of a metal rod that is stationary in 𝑆 . The green lines show lines of simultaneity in each reference frame. Five events are shown on the diagram. For which two events would taking the difference in their 𝑥 -coordinates give the length of the rod in 𝑆 ?

  • A 𝑇 and 𝑊
  • B 𝑃 and 𝑅
  • C 𝑃 and 𝑊
  • D 𝑄 and 𝑇
  • E 𝑃 and 𝑄

Q13:

A spaceship travels from Earth to the Epsilon Eridani system at a speed of 2 × 1 0 m/s relative to Earth. The spaceship measures the distance from Earth to Epsilon Eridani to be 7.75 light-years in its rest frame. What is the distance between Earth and Epsilon Eridani in the rest frame of Earth? Give your answer to 3 significant figures.

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