Special Relativity Theory Foundations
📌 The Special Theory of Relativity was proposed by Albert Einstein in 1905, based on the failure of the Michelson-Morley experiment to detect the hypothetical medium for light, the "ether."
💡 The theory rests on two postulates: 1) The laws of physics are the same in all inertial frames of reference (moving at constant velocity). 2) The speed of light ($c$) in a vacuum is absolute and constant, approximately $3 \times 10^8 \text{ m/s}$, independent of the motion of the source or the observer.
🌠 Consequences of the postulates include the relativity of measurements for length, time, mass, and velocity.

Relativistic Velocity Addition Formula
➕ The formula for adding velocities in special relativity calculates the velocity of object B relative to A ($v_{BA}$) based on their velocities relative to the ground ($t$):
$$v_{BA} = \frac{v_{BT} + v_{TA}}{1 + \frac{v_{BT} v_{TA}}{c^2}}$$
⚖️ Velocities must be treated as vectors (using positive/negative signs for direction); for example, if $v_{B \text{ relative to } T} = -0.6c$, then $v_{T \text{ relative to } B} = +0.6c$.
🚫 The calculated relative velocity must never exceed the speed of light ($c$).

Application Examples (Problem Solving)
🚀 Example 1 Solution: Two rockets, A (moving right at $0.8c$) and B (moving left at $0.6c$) relative to Earth, resulted in the velocity of A observed from B ($v_{AB}$) being $0.946c$.
🚀 Example 2 Solution: Two opposing spacecraft moving away from Earth at speed $v$ (relative to Earth) had a relative speed between them of $1.5v$. Solving the relativistic addition formula showed that the speed $v$ (relative to Earth) was $\frac{1}{3}\sqrt{3}c$, making the final relative speed between the spacecraft $\frac{1}{2}\sqrt{3}c$.
🛰️ Example 3 Solution: A body moving at $0.5c$ relative to Aircraft 1 ($0.8c$) resulted in a calculated velocity of $0.89c$ when measured from Aircraft 2 ($0.2c$), both moving in the same direction relative to Earth.

Key Points & Insights
➡️ Special Relativity, formalized by Einstein in 1905, established the constancy of the speed of light ($c \approx 3 \times 10^8 \text{ m/s}$) as a universal law.
➡️ Mastery of the relativistic velocity addition formula is crucial, requiring careful attention to directional signs when defining velocities relative to a common frame (like Earth/Ground, $t$).
➡️ Always verify that the final calculated relative velocity does not exceed $c$, as this is a fundamental constraint of the theory.

📸 Video summarized with SummaryTube.com on Jan 20, 2026, 06:36 UTC