Special theory of relativity – problems and solutions
1. An object whose length is 60 m moves at a speed of 0.6 c (c = the speed of light). According to a stationary observer, what is the length of the object?
Known :
The proper length (Lo) = 60 meters
Object’s speed (v) = 0.6 c
Wanted : The length contraction (L)
c = the speed of light
Lo = the proper length
L = the length contraction
The length contraction (L) :
2. An event observed by a stationary observer for 12 seconds. What is the time interval of the event, according to an observer who moving at a speed of 0.8 c (c = the speed of light).
Known :
The time interval of the event according to a stationary observer (ts) = 12 seconds
The speed of observer (v) = 0.8 c
Wanted: The time interval of the event according to an observer who moving at a speed of 0.8 c (tm)
Solution :
The equation of the time dilation :
ts = the time interval according to a stationary observer
tm = the time interval according to a moving observer
The time interval according to a moving observer (tm) :
20 conceptual questions and answers related to the special theory of relativity:
1. Question: What fundamental postulates underlie the special theory of relativity?
Answer: The two postulates are: (1) The laws of physics are the same in all inertial frames of reference, and (2) The speed of light in a vacuum is constant and independent of the motion of the source or the observer.
2. Question: How does time dilation occur according to the special theory of relativity?
Answer: Time dilation refers to the phenomenon where time appears to move slower for an observer in motion compared to one at rest, given by the relation Δt’ = Δt/√(1 – v²/c²), where v is the relative velocity and c is the speed of light.
3. Question: What is length contraction?
Answer: Length contraction suggests that an object appears shorter in its direction of motion from the viewpoint of a stationary observer, compared to its proper length when at rest.
4. Question: How does the special theory of relativity affect the concept of simultaneity?
Answer: Events that appear simultaneous to one observer might not be simultaneous to another observer moving at a different velocity. Thus, simultaneity is relative.
5. Question: What is meant by the term “relativistic mass”?
Answer: Relativistic mass refers to the increase in the effective mass of an object as its speed approaches the speed of light. It is given by m = m₀/√(1 – v²/c²), where m₀ is the rest mass.
6. Question: How does the special theory of relativity relate energy and mass?
Answer: The relation is given by E = mc², indicating that energy (E) and mass (m) are equivalent and can be converted into one another.
7. Question: What is the significance of the speed of light (c) in the theory?
Answer: In the special theory of relativity, the speed of light is the maximum speed at which information or matter can travel, and it remains constant in all inertial frames.
8. Question: Why is it impossible for objects with mass to reach or exceed the speed of light?
Answer: As an object’s speed approaches the speed of light, its relativistic mass increases, requiring infinite energy to accelerate it to the speed of light itself.
9. Question: How does velocity addition work in special relativity?
Answer: Unlike classical mechanics, velocities don’t add directly. If u is the speed of one object as observed in a frame moving with speed v, then its speed according to a stationary observer is u’ = (u + v) / (1 + uv/c²).
10. Question: What is the Lorentz transformation?
Answer: Lorentz transformations are mathematical formulas that relate the space and time coordinates of an event in one inertial frame to those in another moving at a constant velocity relative to the first.
11. Question: How does special relativity affect the conservation of momentum?
Answer: Momentum remains conserved, but the definition of momentum becomes relativistic, including the factor of the increased relativistic mass: p = mv/√(1 – v²/c²).
12. Question: Is the special theory of relativity compatible with Newtonian mechanics?
Answer: Yes, for velocities much smaller than the speed of light, the predictions of special relativity converge to those of Newtonian mechanics.
13. Question: What is time dilation’s effect on aging?
Answer: A person traveling at a relativistic speed would age more slowly than someone who remains stationary. This is the “twin paradox” where one twin travels at near-light speed and returns younger than the twin who stayed behind.
14. Question: How does special relativity redefine energy?
Answer: Total energy is the sum of an object’s rest energy (mc²) and its kinetic energy. Even at rest, an object has energy by virtue of its mass.
15. Question: Can anything travel faster than light according to special relativity?
Answer: No, the speed of light is the universal speed limit. Tachyons, hypothetical particles that travel faster than light, aren’t consistent with the theory.
16. Question: How is mass affected at relativistic speeds?
Answer: As an object’s speed approaches the speed of light, its relativistic mass increases without bound, requiring more and more energy for further acceleration.
17. Question: What role does the principle of causality play in special relativity?
Answer: The principle of causality, where cause precedes effect, remains preserved in special relativity, which is why the speed of light limit is crucial.
18. Question: What is spacetime?
Answer: Spacetime combines the three dimensions of space with the fourth dimension of time into a four-dimensional continuum, essential for the framework of relativity.
19. Question: How does special relativity view absolute space and time?
Answer: Special relativity denies the existence of absolute space and time. Both are relative and can vary depending on one’s state of motion.
20. Question: What is an “inertial frame of reference”?
Answer: An inertial frame of reference is one in which objects either remain at rest or move at a constant velocity unless acted upon by an external force. It’s a frame where Newton’s first law of motion holds.
The special theory of relativity, introduced by Albert Einstein, transformed our understanding of space, time, energy, and matter. Its implications extend to many areas of modern physics and have been experimentally verified.
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