Lorentz transformation - time dilation, length contraction, velocity addition law, momentum and energy in relativistic mechanics, centre-of-mass system for two relativistic particles - One Line Questions
1.
The Lorentz factor γ is always greater than or equal to 1. What is its value when v = 0? —
1
2.
What is the relativistic momentum of a photon, which has zero rest mass? —
E/c
3.
If a spaceship moving at 0.9c relative to Earth launches a probe forward at 0.8c relative to the spaceship, what is the speed of the probe relative to Earth, according to the relativistic velocity addition law? —
0.9c + 0.8c = 1.7c / (1 + (0.9c)(0.8c)/c²)
4.
If the velocity of a particle is v = 0.99c, the Lorentz factor γ will be approximately: —
7
5.
Calculate the denominator in the relativistic velocity addition formula for speeds u and v: (u+v)/(denominator). —
1 + uv/c²
6.
If two identical relativistic particles of rest mass m₀ collide head-on and stop in the CMS, what is the total energy in the CMS? —
2m₀c²
7.
In relativistic mechanics, the concept of 'simultaneity' is relative. Two events that are simultaneous in one inertial frame may: —
Not be simultaneous in another inertial frame moving relative to the first.
8.
Which of the following statements about length contraction is correct? —
An object appears shorter to an observer when it moves parallel to its length.
9.
In the context of relativistic mechanics, the term 'proper time' (τ) refers to the time measured by: —
A clock that is stationary relative to the observer measuring it.
10.
In relativistic mechanics, as the velocity v of a particle approaches the speed of light c, its momentum: —
Approaches infinity.
11.
The length contraction formula L = L₀/γ implies that L₀ is the length measured: —
By an observer in the object's rest frame.
12.
From the perspective of the muon itself (in its own rest frame), its lifetime is observed to be: —
Its proper lifetime.
13.
What is the rest energy of a particle with rest mass m₀? —
E = m₀c²
14.
The famous equation E=mc² relates mass and energy. In relativistic mechanics, the total energy E of a particle with rest mass m₀ and velocity v is given by: —
E = γm₀c²
15.
In the centre-of-mass system for two relativistic particles, if particle 1 has energy E₁ and momentum p₁, and particle 2 has energy E₂ and momentum p₂, then: —
E₁ + E₂ = constant, p₁ + p₂ = 0
16.
If a particle has zero rest mass (m₀ = 0), its total energy is given by E = pc. This applies to: —
Photons
17.
The Lorentz transformation is a linear transformation that preserves the spacetime interval Δs² = (cΔt)² - (Δx)² - (Δy)² - (Δz)². This means that special relativity is based on the geometry of: —
Minkowski spacetime
18.
The velocity addition formula ensures that the speed of light c is the ultimate speed limit. If you add any velocity less than c to c, the result is: —
Equal to c
19.
The relativistic addition of velocities is necessary because the Galilean addition law: —
Leads to speeds greater than c when adding velocities close to c.
20.
The concept of 'relativistic mass' (m = γm₀) is useful but sometimes discouraged in modern physics because: —
It obscures the fact that only energy and momentum change with velocity, not rest mass.
21.
Why is the concept of a centre-of-mass system particularly important in relativistic collisions? —
It simplifies calculations of energy and momentum conservation.
22.
The relativistic kinetic energy of a particle is the difference between its total relativistic energy and its rest energy. What is the relativistic kinetic energy (KE)? —
KE = (γ - 1)m₀c²
23.
The relativistic energy-momentum relation states that E² = (pc)² + (m₀c²)². What is E in this equation? —
Total relativistic energy
24.
Consider a rod of length L₀ at rest in a reference frame. What is the length L measured by an observer moving with velocity v relative to the rod? —
L = L₀ / γ
25.
If an observer A sees a rod of length L₀ moving parallel to its length with velocity v, and observer B is at rest with the rod, observer A will measure the rod's length as: —
L₀ / γ
26.
The phenomenon where a moving clock runs slower than an identical clock at rest relative to an observer is known as: —
Time dilation
27.
Consider a particle at rest with mass m₀. Its total energy is E₀ = m₀c². If it is accelerated to a velocity v, its new total energy E will be: —
Greater than E₀
28.
If a particle's speed is v, its relativistic mass is often described as m = γm₀. This 'relativistic mass' concept emphasizes that: —
Mass increases with velocity.
29.
Consider two relativistic particles colliding. The centre-of-mass system (CMS) is defined as the inertial frame in which the total momentum of the system is: —
Zero.
30.
In relativistic mechanics, the energy of a particle is always greater than or equal to its rest energy. The difference is its kinetic energy, which is always: —
Positive
31.
The speed of light 'c' in vacuum is invariant, meaning it is the same for all inertial observers. This is a fundamental postulate of: —
Special relativity
32.
What is the relativistic momentum of a particle with rest mass m₀ and energy E? —
p = E/c
33.
The relativistic momentum of a particle with rest mass m₀ and velocity v is given by: —
p = m₀v / √(1 - v²/c²)
34.
In the centre-of-mass system for two relativistic particles, if the particles have momenta p₁ and p₂, then: —
p₁ = -p₂
35.
The Lorentz contraction affects the length of an object only along the dimension: —
Parallel to its motion.
36.
Length contraction in special relativity occurs along the direction: —
Parallel to the direction of motion.
37.
If a spaceship travels at a significant fraction of the speed of light, an observer on Earth will measure the spaceship's clock to be: —
Running slower than their own clock.
38.
A muon created in the upper atmosphere travels towards Earth at a speed close to the speed of light. Due to time dilation, an observer on Earth observes the muon's lifetime to be: —
Longer than its proper lifetime.
39.
When considering the centre-of-mass system for two relativistic particles, the total energy in this frame is often denoted by W. If the particles are created from rest and move apart, W represents the: —
Sum of their rest energies.
40.
What is the term 'γ' (gamma) in the Lorentz transformation equations, where v is the relative velocity between frames and c is the speed of light? —
The Lorentz factor, given by 1/√(1 - v²/c²)
41.
The time dilation formula Δt = γΔt₀ implies that Δt is the time interval measured by: —
An observer in a frame moving relative to the clock.
42.
In the centre-of-mass system of two particles, the total kinetic energy is minimized when: —
The particles have zero momentum.
43.
The classical Galilean velocity addition states that if observer A sees object P moving at speed u, and observer B moves past A at speed v, then B sees P moving at speed u+v. The relativistic velocity addition law differs when: —
The speeds are comparable to the speed of light.
44.
Which of the following is NOT a consequence of special relativity? —
Gravitational lensing
45.
Which of the following physical quantities is invariant under Lorentz transformations? —
Spacetime interval
46.
What is the primary purpose of the Lorentz transformation equations in special relativity? —
To relate the space and time coordinates of an event as measured by two observers in inertial frames moving at a constant velocity relative to each other.
47.
According to the relativistic velocity addition law, if two objects are moving towards each other with speeds u and v relative to a third frame, and u and v are close to c, their relative speed is: —
(u + v) / (1 + uv/c²)
48.
The Lorentz transformation equations reduce to the Galilean transformation equations under which condition? —
When v is much less than c.
49.
If an event occurs at (x, t) in one inertial frame, what are its coordinates (x', t') in a frame moving with velocity v along the x-axis relative to the first frame, according to Lorentz transformation? —
x' = γ(x - vt), t' = γ(t - vx/c²)
50.
In the Lorentz transformation, if an observer is at rest and measures a time interval Δt, what is the time interval Δt' measured by an observer moving with velocity v relative to the first observer? —
Δt' = Δt / γ