Worksheet: World Lines in Minkowski Diagrams

In this worksheet, we will practice interpreting the world lines of objects in Minkowski diagrams.

Q1:

The following Minkowski diagrams show the world lines of different particles in an inertial frame ๐‘† . Four of the diagrams show only hypothetical world lines, as they show a particle that for part or all of its world line moves faster than the speed of light. Which diagram shows the world line of a particle that at no point moves faster than the speed of light?

  • A
  • B
  • C
  • D
  • E

Q2:

The following Minkowski diagrams show the world lines of different particles in an inertial frame ๐‘† . Which diagram shows a particle moving faster than the speed of light?

  • A
  • B
  • C
  • D
  • E

Q3:

The following Minkowski diagrams show the world lines of different particles in an inertial frame ๐‘† . Which diagram shows a particle that does not move at a constant speed?

  • A
  • B
  • C
  • D
  • E

Q4:

The following Minkowski diagrams show the world lines of different particles in an inertial frame ๐‘† . Which diagram shows a particle that moves at a constant nonzero speed in ๐‘† for the entirety of its world line?

  • A
  • B
  • C
  • D
  • E

Q5:

The following Minkowski diagrams show two inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก and frame ๐‘† ๏Ž˜ with axes ๐‘ฅ ๏Ž˜ and ๐‘ ๐‘ก ๏Ž˜ . In which diagram is the red line constant in time in frame ๐‘† ๏Ž˜ ?

  • A
  • B
  • C
  • D
  • E

Q6:

The following Minkowski diagrams show two inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก and frame ๐‘† ๏Ž˜ with axes ๐‘ฅ ๏Ž˜ and ๐‘ ๐‘ก ๏Ž˜ . The red line shows the world line of a particle. Which diagram shows a particle that is not moving at a constant speed in ๐‘† ๏Ž˜ ?

  • A
  • B
  • C
  • D
  • E

Q7:

The following Minkowski diagrams show the world lines of different particles in an inertial frame ๐‘† . Which diagram shows a particle that is at rest in ๐‘† ?

  • A
  • B
  • C
  • D

Q8:

The following Minkowski diagrams show an inertial frame ๐‘† . In which diagram is the red line constant in time in frame ๐‘† ?

  • A
  • B
  • C
  • D
  • E

Q9:

The given Minkowski diagram shows two inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก and frame ๐‘† ๏Ž˜ with axes ๐‘ฅ ๏Ž˜ and ๐‘ ๐‘ก ๏Ž˜ . Which two of the following does the red line in the diagram show?

  1. The world line of an object that is moving at a constant speed in ๐‘†
  2. The world line of an object that is at rest in frame ๐‘† ๏Ž˜
  3. The world line of an object that is moving at the speed of light, ๐‘ , in both ๐‘† and ๐‘† ๏Ž˜
  4. The world line of an object that is at rest in frame ๐‘†
  5. A line of constant time in ๐‘†
  6. A line of constant time in ๐‘† ๏Ž˜
  7. The world line of an object that is moving at a nonconstant speed in both ๐‘† and ๐‘† ๏Ž˜
  8. The world line of an object that is moving at a constant speed in ๐‘† ๏Ž˜
  • Ai and ii
  • Biv and viii
  • Ciii and iv
  • Dv and vi
  • Eviii and vi

Q10:

The Minkowski diagram shows three inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก , frame ๐‘† โ€ฒ with axes ๐‘ฅ โ€ฒ and ๐‘ ๐‘ก โ€ฒ , and frame ๐‘† โ€ฒ โ€ฒ with axes ๐‘ฅ โ€ฒ โ€ฒ and ๐‘ ๐‘ก โ€ฒ โ€ฒ . The red line shows the world line of a particle.

What is the speed of the particle in frame ๐‘† โ€ฒ โ€ฒ ?

  • A ๐‘
  • B ๐‘ 3
  • C ๐‘ โˆš 2
  • D ๐‘ 4
  • E ๐‘ 2

What is the speed of the particle in frame ๐‘† โ€ฒ ?

  • A ๐‘
  • B ๐‘ 3
  • C ๐‘ โˆš 2
  • D ๐‘ 4
  • E ๐‘ 2

Q11:

The Minkowski diagram shows three inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก , frame ๐‘† โ€ฒ with axes ๐‘ฅ โ€ฒ and ๐‘ ๐‘ก โ€ฒ , and frame ๐‘† โ€ฒ โ€ฒ with axes ๐‘ฅ โ€ฒ โ€ฒ and ๐‘ ๐‘ก โ€ฒ โ€ฒ . The red line shows the world line of a particle. In which inertial frame is the particle at rest?

  • A ๐‘† โ€ฒ
  • B ๐‘†
  • C ๐‘† โ€ฒ โ€ฒ

Q12:

The Minkowski diagram shows three inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก , frame ๐‘† โ€ฒ with axes ๐‘ฅ โ€ฒ and ๐‘ ๐‘ก โ€ฒ , and frame ๐‘† โ€ฒ โ€ฒ with axes ๐‘ฅ โ€ฒ โ€ฒ and ๐‘ ๐‘ก โ€ฒ โ€ฒ . The red line shows the world line of a particle.

In which inertial frame does the particle have the greatest speed?

  • A ๐‘† โ€ฒ โ€ฒ
  • B ๐‘†
  • C ๐‘† โ€ฒ

In which inertial frame is the particle at rest?

  • A ๐‘†
  • B ๐‘† โ€ฒ
  • C ๐‘† โ€ฒ โ€ฒ

Q13:

The above Minkowski diagram shows three inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก , frame ๐‘† โ€ฒ with axes ๐‘ฅ โ€ฒ and ๐‘ ๐‘ก โ€ฒ , and frame ๐‘† โ€ฒ โ€ฒ with axes ๐‘ฅ โ€ฒ โ€ฒ and ๐‘ ๐‘ก โ€ฒ โ€ฒ . Which frame has the greatest speed relative to ๐‘† ?

  • A ๐‘† โ€ฒ โ€ฒ
  • B ๐‘† โ€ฒ

Q14:

The given Minkowski diagram shows two inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก and frame ๐‘† โ€ฒ with axes ๐‘ฅ โ€ฒ and ๐‘ ๐‘ก โ€ฒ . Which two of the following does the red line in the diagram show?

  1. A line of constant time in ๐‘†
  2. The world line of an object that is moving at the speed of light, ๐‘ , in both ๐‘† and ๐‘† โ€ฒ
  3. The world line of an object that is at rest in frame ๐‘†
  4. The world line of an object that is moving at a constant speed in ๐‘† โ€ฒ
  5. A line of constant time in ๐‘† โ€ฒ
  6. The world line of an object that is moving at a constant speed in ๐‘†
  7. The world line of an object that is moving at a nonconstant speed in both ๐‘† and ๐‘† โ€ฒ
  8. The world line of an object that is at rest in frame ๐‘† โ€ฒ
  • Aviii and v
  • Biii and vi
  • Civ and vi
  • Di and ii
  • Eiv and vii

Q15:

The Minkowski diagram shows three inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก , frame ๐‘† โ€ฒ with axes ๐‘ฅ โ€ฒ and ๐‘ ๐‘ก โ€ฒ , and frame ๐‘† โ€ฒ โ€ฒ with axes ๐‘ฅ โ€ฒ โ€ฒ and ๐‘ ๐‘ก โ€ฒ โ€ฒ . The red line shows the world line of a particle. In which inertial frame does the particle have the greatest speed?

  • A ๐‘† โ€ฒ โ€ฒ
  • B ๐‘† โ€ฒ
  • C ๐‘†

Q16:

The Minkowski diagram shows two inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก and frame ๐‘† โ€ฒ with axes ๐‘ฅ โ€ฒ and ๐‘ ๐‘ก โ€ฒ . Which of the following does the red line in the diagram show?

  1. A line of constant time in ๐‘†
  2. The world line of an object that is moving at the speed of light, ๐‘ , in both ๐‘† and ๐‘† โ€ฒ
  3. The world line of an object that is at rest in frame ๐‘†
  4. The world line of an object that is moving at a constant speed in ๐‘† โ€ฒ
  5. A line of constant time in ๐‘† โ€ฒ
  6. The world line of an object that is moving at a constant speed in ๐‘†
  7. The world line of an object that is moving at a nonconstant speed in both ๐‘† and ๐‘† โ€ฒ
  8. The world line of an object that is at rest in frame ๐‘† โ€ฒ
  • Aiv
  • Bii
  • Cvii
  • Dvi
  • Eiii

Q17:

The following Minkowski diagrams show two inertial frames: frame ๐‘† with axes ๐‘ฅ and ๐‘ ๐‘ก and frame ๐‘† ๏Ž˜ with axes ๐‘ฅ ๏Ž˜ and ๐‘ ๐‘ก ๏Ž˜ . The red line shows the world line of a particle. Which diagram shows a particle that is at rest in the frame ๐‘† ๏Ž˜ ?

  • A
  • B
  • C
  • D
  • E

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