Many very good HSC Physics students become confused in the relativity section of the space topic due to the poor wording of the practice questions in some textbooks. Here is a graded set of tutorial questions to help students gain confidence in time dilation problems. The questions refer to hypothetical objects and are designed for examination purposes. These questions are numerical, so that students can gain an understanding of these concepts by attaching numbers to them. The answers are given in square brackets after the question.
Time. These questions compare the time intervals to (the time interval of an event as measured on a clock in the reference frame in which the event happens at the same location, called the proper time) and tv (the time interval of the event according to a clock in any other inertial reference frame) using the equation
tv = to/√(1 - v2/c2)
"Event" means something happening in one of the reference frames, such as an experiment being carried out at the same position in a spaceship or the decay of a very rapidly moving charged particle, such as a muon moving through the earth's atmosphere. The reference frame in which the event occurs at the same location is called the rest reference frame and proper time is the time measured by a clock in this reference frame. In the muon example, the rest frame is the frame in which the muon is at rest, and this is the frame moving with the muon. Note that tv is always greater than to and so this effect is called time dilation. We do not take in to account the travel time of the light rays travelling between the reference frames. For example, if an experiment is performed in a high speed speed rocket moving away from the earth the light rays from the end position of the experiment will have a greater distance to travel than those coming from the starting position and so this would add extra time. The confusion in time dilation problems arises due to the fact that in Einstein's theory of special relativity compared time intervals depend on position (4 dimensional space-time) and the location of the event that is being measured must be given so that the rest reference frame in which the proper time occurs can be identified.
A charged particle has a lifetime of 5.0 ms when measured from its own reference frame. The particle moves at 2.7x108 m/s relative to a laboratory. What is the lifetime of the particle according to a person in the laboratory? [11 ms]
A subatomic particle moves at 0.92c along the tube of a linear accelerator. Scientists working in the laboratory measure that the particle decays after 30 ms. What is the lifetime of the particle in its own reference frame?[12 ms]
A particle has a lifetime of 12 ns when measured in its rest frame. Scientists working in a laboratory measure the lifetime of the particle as 18 ns. What is the speed of the particle relative to the laboratory?[0.75c]
A rocket moves away from the earth at a constant relative speed of 0.85c. The crew of the rocket perform an experiment that lasts for a time interval of 3.0 minutes according to their own clocks. What is the time interval of the experiment according to clocks on the earth? [5.7 minutes]
- Two spaceships, X and Y, move in the same direction in the same straight line. The relative speed of the spaceships is constant at 0.96c. The crew of Y perform an experiment for a time interval of 14 s according to their own clocks. What is the time duration of the experiment according to clocks of spaceship X? [50 s]
- A subatomic particle approaches the earth at 5.7x107 m/s. Scientists in the laboratory on earth measure that the particle was moving for 51 ms. What is the lifetime of the particle in its own reference frame? [50 ms]
- A spaceship moves away from the earth at a constant relative speed of 0.86c. Observers on the earth measure that an experiment performed on the spaceship lasts for a duration of 37 minutes. What is the duration of the experiment according to the crew of the spaceship? [19 minutes]
- A spaceship moves at a constant speed of 0.98c relative to the earth. A clock on the spaceship measures that an on-board experiment lasts for 6.0 hours. What is the time interval of this experiment according to observers on the earth? [30 hours]
- A spaceship moves at a constant speed of 0.98c relative to the earth. A clock on the earth measures that the time interval for an experiment performed on the spaceship is 6.0 hours. What is the time duration of this experiment as shown by a clock on the spaceship? [1.2 hours]
- A spaceship moves towards the earth at a relative speed of 0.90c. Scientists in a laboratory on earth do an experiment that lasts for one day according to their clocks. What is the time interval of this experiment according to a clock on the spaceship? [2.3 days]
- Two spaceships, P and Q, move towards each other in the same straight line at a relative speed of 0.94c. The crew of P do an experiment that takes 6 hours according to a clock in Q. What is the time taken to do the experiment according to the crew of P? [2 hours]
- A spaceship moves away from the earth at 0.99c. A clock on the earth shows a time interval of 5.0 hours. What is the time interval shown on the spaceship clock? [This is an example of an ambiguous question often seen in texts. Which is the proper time? The question needs to state the rest reference frame of the event that is measured by these time intervals so that the proper time can be identified]