The Concept of Relativistic Mass
In classical physics, mass is considered a constant property of an object. However, Einstein's theory of special relativity shows that mass is not constant, but increases with velocity. This increase becomes significant as an object approaches the speed of light.
The Formula
The relativistic mass (m) of a particle is given by:
m = m₀ / √(1 - v²/c²)
where:
* m₀ is the rest mass (mass at rest)
* v is the velocity of the particle
* c is the speed of light
The Challenge
You want to find the speed (v) at which the relativistic mass (m) is double the rest mass (m₀). So, we set m = 2m₀ and solve for v:
2m₀ = m₀ / √(1 - v²/c²)
Solving for v
1. Divide both sides by m₀: 2 = 1 / √(1 - v²/c²)
2. Square both sides: 4 = 1 / (1 - v²/c²)
3. Take the reciprocal of both sides: 1/4 = 1 - v²/c²
4. Rearrange: v²/c² = 3/4
5. Take the square root of both sides: v/c = √(3/4)
6. Solve for v: v = c * √(3/4) ≈ 0.866c
Conclusion
A particle must move at approximately 86.6% the speed of light for its relativistic mass to double.
Important Note: It is not possible for a particle to reach the speed of light (c). This is because as the particle approaches the speed of light, its relativistic mass approaches infinity, requiring an infinite amount of energy to accelerate it further.
