Worksheet: Events in Minkowski Diagrams

In this worksheet, we will practice determining the order of events as perceived from different inertial reference frames and determining whether events are simultaneous.

Q1:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Four events are shown in the diagram: 𝑃 , 𝑄 , 𝑅 , and 𝑇 .

List the events in order of their distance from the origin in frame 𝑆 , starting with the closest.

  • A 𝑄 , 𝑃 , 𝑅 , 𝑇
  • B 𝑅 , 𝑃 , 𝑇 , 𝑄
  • C 𝑇 , 𝑄 , 𝑅 , 𝑃
  • D 𝑃 , 𝑅 , 𝑇 , 𝑄
  • E 𝑃 , 𝑅 , 𝑄 , 𝑇

List the events in order of their distance from the origin in frame 𝑆 , starting with the closest.

  • A 𝑃 , 𝑄 , 𝑇 , 𝑅
  • B 𝑇 , 𝑄 , 𝑃 , 𝑅
  • C 𝑅 , 𝑃 , 𝑇 , 𝑄
  • D 𝑃 , 𝑅 , 𝑄 , 𝑇
  • E 𝑇 , 𝑄 , 𝑅 , 𝑃

Q2:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Six events are shown on the diagram: 𝑃 , 𝑄 , 𝑅 , 𝑇 , 𝑊 , and 𝑍 .

Which two events occur at the same distance away from the origin in frame 𝑆 ?

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

Which two events occur at the same distance away from the origin in frame 𝑆 ?

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

Q3:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Three events are shown on the diagram: 𝑃 , 𝑄 , and 𝑅 .

In which frame do all three events occur at the same location?

  • A 𝑆
  • B 𝑆

Which event occurs farthest from the origin in frame 𝑆 ?

  • A 𝑃
  • B 𝑄
  • C 𝑅

Q4:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Three events are shown on the diagram: 𝑃 , 𝑄 , and 𝑅 . In frame 𝑆 , all three events occur in the same location. Which event occurs closest to the origin in frame 𝑆 ?

  • A 𝑄
  • B 𝑃
  • C 𝑅

Q5:

The Minkowski diagram shows four events, 𝑃 , 𝑄 , 𝑅 , and 𝑇 , in the inertial frame 𝑆 . Each event represents a light bulb flashing on for a fraction of a second. The light emitted from each bulb then travels at speed 𝑐 . For an observer in frame 𝑆 at position 𝑥 = 0 , the light from two events will reach 𝑥 = 0 at the same time; thus, the observer at 𝑥 = 0 will see the two events happen at the same time. Which two events will the observer see happen at the same time?

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

Q6:

The Minkowski diagram shows two inertial frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed 𝑢 relative to frame 𝑆 . Three events are marked on the diagram: 𝑄 , 𝑅 , and 𝑇 . In frame 𝑆 , which event occurs first?

  • A 𝑇
  • B 𝑄
  • C 𝑅

Q7:

The Minkowski diagram shows two inertial frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed 𝑢 relative to frame 𝑆 . Each of the events 𝑄 , 𝑅 , and 𝑇 represents a light bulb flashing on for a fraction of a second. The light is emitted from each bulb and then travels at speed 𝑐 . If there is an observer at rest in frame 𝑆 , which light bulb will that observer see flash first?

  • A 𝑇
  • B 𝑄
  • C 𝑅

Q8:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Six events are shown on the diagram: 𝑃 , 𝑄 , 𝑅 , 𝑇 , 𝑊 , and 𝑍 .

Which two events are simultaneous in the frame 𝑆 ?

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

Which two events are simultaneous in the frame 𝑆 ?

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

Q9:

The given Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Two events are shown in the diagram: 𝑃 and 𝑄 . In both frames 𝑆 and 𝑆 , the dotted lines are spaced 1 m apart along the 𝑥 -axis.

What is the spatial separation of events 𝑃 and 𝑄 in frame 𝑆 ?

If Δ 𝑥 is the separation of the events in frame 𝑆 and Δ 𝑥 is the separation of the events in frame 𝑆 , which of the following expressions relating these two quantities is true?

  • A Δ 𝑥 = 1 2 Δ 𝑥
  • B Δ 𝑥 = Δ 𝑥
  • C Δ 𝑥 = 2 Δ 𝑥
  • D Δ 𝑥 < Δ 𝑥
  • E Δ 𝑥 > Δ 𝑥

Q10:

The given Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Frame 𝑆 moves at a speed 𝑢 relative to 𝑆 . Two events, 𝑄 and 𝑅 , are marked on the diagram.

In which frame do the two events occur at the same time?

  • A 𝑆
  • B 𝑆

Which event occurs first in frame 𝑆 ?

  • A 𝑅
  • BThey occur at the same time.
  • C 𝑄

Q11:

The Minkowski diagram shows four events, 𝑃 , 𝑄 , 𝑅 , and 𝑇 , in the inertial frame 𝑆 . For an observer in frame 𝑆 at the position 𝑥 = 0 , which event occurs closest to the observer?

  • A 𝑄
  • B 𝑅
  • C 𝑇
  • D 𝑃

Q12:

The above Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Four events are shown on the diagram: 𝑃 , 𝑄 , 𝑅 , and 𝑇 .

List the events in order of their distance from the origin in frame 𝑆 , starting with the closest.

  • AP, 𝑄 , 𝑅 , 𝑇
  • B 𝑇 , 𝑃 , 𝑄 , 𝑅
  • C 𝑄 , 𝑅 , 𝑃 , 𝑇
  • D 𝑇 , 𝑄 , 𝑃 , 𝑅
  • E 𝑅 , 𝑄 , 𝑃 , 𝑇

List the events in order of their distance from the origin in frame 𝑆 , starting with the closest.

  • A 𝑇 , 𝑄 , 𝑃 , 𝑅
  • B 𝑅 , 𝑄 , 𝑇 , 𝑃
  • C 𝑃 , 𝑇 , 𝑅 , 𝑄
  • D 𝑄 , 𝑇 , 𝑃 , 𝑅
  • E 𝑃 , 𝑇 , 𝑄 , 𝑅

Q13:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Four events are shown on the diagram: 𝑃 , 𝑄 , 𝑅 , and 𝑇 .

In frame 𝑆 , in what order do the events occur?

  • A 𝑃 , 𝑄 , 𝑅 , 𝑇
  • B 𝑃 , 𝑄 , 𝑇 , 𝑅
  • C 𝑃 , 𝑅 , 𝑄 , 𝑇
  • D 𝑇 , 𝑄 , 𝑅 , 𝑃
  • E 𝑇 , 𝑅 , 𝑄 , 𝑃

In frame 𝑆 , in what order do the events occur?

  • A 𝑇 , 𝑄 , 𝑅 , 𝑃
  • B 𝑃 , 𝑄 , 𝑇 , 𝑅
  • C 𝑃 , 𝑅 , 𝑄 , 𝑇
  • D 𝑄 , 𝑇 , 𝑅 , 𝑃
  • E 𝑃 , 𝑄 , 𝑅 , 𝑇

Q14:

The Minkowski diagram shows four events, 𝑃 , 𝑄 , 𝑅 , and 𝑇 , in the inertial frame 𝑆 . Which event occurs first in frame 𝑆 ?

  • A 𝑅
  • B 𝑇
  • C 𝑄
  • D 𝑃

Q15:

The above Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐 𝑡 and frame 𝑆 with axes 𝑥 and 𝑐 𝑡 . Four events are shown on the diagram: 𝑃 , 𝑄 , 𝑅 , and 𝑇 .

In frame 𝑆 , in what order do the events occur?

  • A 𝑃 , 𝑅 , 𝑇 , 𝑄
  • B 𝑅 , 𝑄 , 𝑇 , 𝑅
  • C 𝑇 , 𝑅 , 𝑄 , 𝑃
  • D 𝑄 , 𝑃 , 𝑅 , 𝑇
  • E 𝑄 , 𝑇 , 𝑃 , 𝑅

In frame 𝑆 , in what order do the events occur?

  • A 𝑄 , 𝑃 , 𝑅 , 𝑇
  • B 𝑇 , 𝑅 , 𝑃 , 𝑄
  • C 𝑇 , 𝑃 , 𝑄 , 𝑅
  • D 𝑃 , 𝑅 , 𝑇 , 𝑄
  • E 𝑅 , 𝑇 , 𝑄 , 𝑇

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