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 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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.

  • A𝑅, 𝑃, 𝑇, 𝑄
  • B𝑄, 𝑃, 𝑅, 𝑇
  • C𝑃, 𝑅, 𝑇, 𝑄
  • D𝑇, 𝑄, 𝑅, 𝑃
  • E𝑃, 𝑅, 𝑄, 𝑇

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

  • A𝑇, 𝑄, 𝑅, 𝑃
  • B𝑅, 𝑃, 𝑇, 𝑄
  • C𝑃, 𝑅, 𝑄, 𝑇
  • D𝑇, 𝑄, 𝑃, 𝑅
  • E𝑃, 𝑄, 𝑇, 𝑅

Q2:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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?

  • A𝑍 and 𝑃
  • B𝑅 and 𝑍
  • C𝑄 and 𝑇
  • D𝑅 and 𝑊
  • E𝑍 and 𝑊

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

  • A𝑄 and 𝑇
  • B𝑅 and 𝑍
  • C𝑅 and 𝑊
  • D𝑍 and 𝑊
  • E𝑍 and 𝑃

Q3:

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

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

  • A𝑆
  • B𝑆

Which event occurs farthest from the origin in frame 𝑆?

  • A𝑄
  • B𝑅
  • C𝑃

Q4:

The Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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?

  • A𝑅
  • B𝑃
  • C𝑄

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?

  • A𝑃 and 𝑅
  • B𝑇 and 𝑅
  • C𝑄 and 𝑃
  • D𝑄 and 𝑇
  • E𝑃 and 𝑇

Q6:

The Minkowski diagram shows two inertial frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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?

  • A𝑄
  • B𝑇
  • C𝑅

Q7:

The Minkowski diagram shows two inertial frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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?

  • A𝑄
  • B𝑅
  • C𝑇

Q8:

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

Which two events are simultaneous in the frame 𝑆?

  • A𝑅 and 𝑇
  • B𝑊 and 𝑍
  • C𝑃 and 𝑅
  • D𝑍 and 𝑇
  • E𝑄 and 𝑊

Which two events are simultaneous in the frame 𝑆?

  • A𝑍 and 𝑇
  • B𝑅 and 𝑇
  • C𝑊 and 𝑍
  • D𝑄 and 𝑊
  • E𝑃 and 𝑅

Q9:

The given Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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 Δ𝑥 is the separation of the events in frame 𝑆 and Δ𝑥 is the separation of the events in frame 𝑆, which of the following expressions relating these two quantities is true?

  • AΔ𝑥=2Δ𝑥
  • BΔ𝑥=12Δ𝑥
  • CΔ𝑥<Δ𝑥
  • DΔ𝑥>Δ𝑥
  • EΔ𝑥=Δ𝑥

Q10:

The given Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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?

  • A𝑆
  • B𝑆

Which event occurs first in frame 𝑆?

  • AThey occur at the same time.
  • B𝑄
  • C𝑅

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?

  • A𝑅
  • B𝑄
  • C𝑃
  • D𝑇

Q12:

The above Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. 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.

  • A𝑇, 𝑄, 𝑃, 𝑅
  • B𝑇, 𝑃, 𝑄, 𝑅
  • CP, 𝑄, 𝑅, 𝑇
  • D𝑄, 𝑅, 𝑃, 𝑇
  • E𝑅, 𝑄, 𝑃, 𝑇

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

  • A𝑅, 𝑄, 𝑇, 𝑃
  • B𝑃, 𝑇, 𝑅, 𝑄
  • C𝑇, 𝑄, 𝑃, 𝑅
  • D𝑃, 𝑇, 𝑄, 𝑅
  • E𝑄, 𝑇, 𝑃, 𝑅

Q13:

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

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

  • A𝑃, 𝑅, 𝑄, 𝑇
  • B𝑇, 𝑄, 𝑅, 𝑃
  • C𝑃, 𝑄, 𝑅, 𝑇
  • D𝑃, 𝑄, 𝑇, 𝑅
  • E𝑇, 𝑅, 𝑄, 𝑃

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

  • A𝑃, 𝑅, 𝑄, 𝑇
  • B𝑃, 𝑄, 𝑅, 𝑇
  • C𝑄, 𝑇, 𝑅, 𝑃
  • D𝑃, 𝑄, 𝑇, 𝑅
  • E𝑇, 𝑄, 𝑅, 𝑃

Q14:

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

  • A𝑇
  • B𝑅
  • C𝑃
  • D𝑄

Q15:

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

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

  • A𝑄, 𝑃, 𝑅, 𝑇
  • B𝑄, 𝑇, 𝑃, 𝑅
  • C𝑃, 𝑅, 𝑇, 𝑄
  • D𝑅, 𝑄, 𝑇, 𝑅
  • E𝑇, 𝑅, 𝑄, 𝑃

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

  • A𝑇, 𝑅, 𝑃, 𝑄
  • B𝑅, 𝑇, 𝑄, 𝑇
  • C𝑃, 𝑅, 𝑇, 𝑄
  • D𝑄, 𝑃, 𝑅, 𝑇
  • E𝑇, 𝑃, 𝑄, 𝑅

Q16:

The given Minkowski diagram shows two frames: frame 𝑆 with axes 𝑥 and 𝑐𝑡 and frame 𝑆 with axes 𝑥 and 𝑐𝑡. Two events are shown on the diagram: 𝑃 and 𝑄. In both frames 𝑆 and 𝑆, the dotted lines are spaced 1 light-second apart along the 𝑦-axis.

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

  • A4 light-seconds
  • B1 light-second
  • C0.5 light-seconds
  • D1.5 light-seconds
  • E2 light-seconds

If Δ𝑡 is the time interval of the events in frame 𝑆 and Δ𝑡 is the time interval of the events in frame 𝑆, which of the following expressions relating these two quantities is true?

  • AΔ𝑡>Δ𝑡
  • BΔ𝑡=2Δ𝑡
  • CΔ𝑡=12Δ𝑡
  • DΔ𝑡<Δ𝑡
  • EΔ𝑡=Δ𝑡

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