Unit 11: Special Theory of Relativity — Short Questions

See Q.2 of theory.

See Q.4 of theory.

According to special relativity, a material particle can't reach the speed of light because its mass would increase
As m = m₀ / √(1 - v²/c²)
Assuming speed of material object equal to speed of light i.e v = c then
m = m₀ / √(1 - c²/c²) = m₀ / √(1 - 1) = ∞
So, the energy required would become infinite. This makes it physically impossible, because an infinite force is required to accelerate it, as the infinite force is not available. So a material object cannot accelerate to the speed of light in free space.

The theory of relativity agrees with Newton's Laws of motion under limitation on speed. At low speed, Newton's laws work well and give accurate results but for bodies moving at low speed as compared to speed of light, the relativistic results are negligible small.
When we deal with atomic particles moving with velocities approaching speed of light, the experimental results cannot be explained without taking Einstein's equation into account.

Proper time Time measured by a clock at rest relative to the observer/event.
Proper length: Length of an object measured by an observer at rest relative to the object.

Relativistic Concepts
Relativistic Mass
Relativistic mass refers to the increase in mass of an object as its speed approaches the speed of light, according to special relativity.
Mathematically: m = m₀ / √(1 - v²/c²)
Here m₀ is rest mass and 'm' is relativistic mass.

Relativistic Length (Length Contraction)
Relativistic length contraction refers to the shortening of an object's length as observed by an observer in motion relative to the object.
Mathematically: ℓ = ℓ₀√(1 - v²/c²)
Hear ℓ is proper length and c is relativistic length.

Relativistic Time (Time Dilation)
Relativistic time dilation refers to the slowing down of time as observed by an observer in motion relative to a stationary observer or clock.
Mathematically: t = t₀ / √(1 - v²/c²)
Here 'to' is proper time and 't' is relativistic time.

According to Einstein's special theory of relativity, when an object moves at a speed close to the speed of light, its inertia increases. This means it becomes harder to accelerate the object. This effect is observed as an increase in mass, called relativistic mass.
The formula is:
m = m₀ / √(1 - v²/c²)
• As speed v increases, the denominator becomes smaller, so the mass m becomes larger.

Space-time is not absolute. It depends upon the state of observer. It can stretch or bend due to motion or gravity.
When there is no motion or change in gravity. The space-time remains flat and unchanged.

As an object moves faster and approaches the speed of light, its relativistic mass increases according to
the formula: m = m₀ / √(1 - v²/c²)
As v approaches c (speed of light)
v → c ⟹ v/c → 1
⟹ m → ∞
As infinite mass would require an infinite force to accelerate it. Since infinite forces are not available, hence no material object can reach or exceed the speed of light, making it the ultimate speed limit in nature.

Special relativity has been confirmed through various experiments and applications, such as:
(i) Particle accelerators:
Particles gain relativistic mass when moving at speeds close to the speed of light.
(ii) GPS system:
Time dilation occurs in GPS satellites due to their high speed and altitude. These time differences match predictions from relativity and require correction.
(iii) Nuclear reactions:
The equation E = mc² is verified in nuclear reactions where small amounts of mass convert into large amounts of energy.
e.g. fission reaction, Fusion reaction, pair production and transmutation etc.

The main consequences of Einstein's theory are:
(i) The mass of the body increases with increase in velocity decreases.
(ii) The length of a body along the direction of motion decreases when it moves with the speed comparable to the speed of light.
(iii) A clock running at the speed comparable to the speed of light will run slower than any other ordinary clock (called time dilation)

Let "ℓ" is the length of a spacecraft moving with the speed of light i-e 3 x 10⁸ m/s then according to Einstein's the length can be determined by the following equation.
ℓ = ℓ₀√(1 - v²/c²)
If v = c
ℓ = ℓ₀√(1 - (c/c)²)
l = 0

By Einstein's the relativistic mass "m" can be determined by the helps of the following equation.
m = m₀ / √(1 - (v/c)²)
m = m₀ / √(1 - (0.999c)²/c²)
m = m₀ / √(1 - (0.999)² × c²/c²)
m = m₀ / √(1 - (0.999)² × c²/c²)
m = 22.4 m₀

By Einstein's energy and mass are interconvertable i.e.
E = mc²
E = 1 × (3 × 10⁸)²
E = 9 × 10¹⁶ J

We know that mass 'm' of an object moving at a speed 'v' comparable to the speed of light is different as compared to the rest mass 'm₀' i.e.
m = m₀ / √(1 - v²/c²)
If v = c then m → 0, this means an infinite force is required to accelerate a body to a speed equal to the speed of light.
As an infinite force is not available therefore no material object can be accelerated to a speed equal to the speed of light.

No, the speed of light is not affected by relative motion between the observer and source. According to special relativity, the speed of light remains constant for all observers, regardless of their relative motion. This is a fundamental difference between light and other waves, like sound.

According to relativistic equation
m = m₀ / √(1 - v²/c²). As v = c so, m = m₀ / √(1 - c²/c²)
m = m₀ / √(0) = ∞
Since infinite mass requires an infinite force to accelerate the object, which is not possible physically so a material object like a star cannot move with a velocity equal to the velocity light

If the speed of light were 50 ms⁻¹
Communication and visual feedback would be noticeably delayed.

See Q.2 of theory.

See Q.4 of theory.

See Q.5 of theory.