A physicist applies a constant force to an object already moving at 0.99c. Compared to applying the same force to an identical object at rest, the resulting acceleration is:
AThe same — Newton's second law F = ma holds in all inertial frames regardless of speed
BZero — no force can produce acceleration in an object moving at near-light speed
CMuch smaller — the object's effective inertia has grown by a factor of γ ≈ 7, so each unit of force produces far less acceleration
DLarger — the force acts on a faster-moving object and therefore delivers more kinetic energy per unit time
Relativistic momentum p = γmv means the effective inertia resisting further acceleration is γm, not just m. At 0.99c, γ ≈ 7, so the object resists acceleration about seven times more strongly than at rest. Each additional increment of speed costs far more momentum (and energy), and those increments keep shrinking. This is not a technological limitation — it is built into the structure of relativistic dynamics.
Question 2 Multiple Choice
Why is p = γmv defined as relativistic momentum rather than the classical p = mv?
ABecause γmv approaches infinity near c, preventing massive objects from reaching light speed
BBecause γmv reduces to mv at low speeds, providing the correct classical limit
CBecause γmv is the quantity conserved in all inertial frames connected by Lorentz transformations, while mv is not — it is the Lorentz-invariant definition that preserves momentum conservation across frames
DBecause Einstein derived it directly from the mass-energy relation E = mc²
The conceptual reason for the definition is conservation. Consider a symmetric collision analyzed from two inertial frames related by a Lorentz boost. Classical momentum mv is NOT conserved in both frames simultaneously. Relativistic momentum p = γmv IS. This is what singles out γmv as the correct quantity — it transforms consistently under Lorentz transformations, preserving momentum conservation in every inertial frame. The infinity-at-c consequence follows from the definition; it is not the reason for it.
Question 3 True / False
Near the speed of light, a particle's rest mass increases, which is why it becomes increasingly difficult to accelerate further.
TTrue
FFalse
Answer: False
Rest mass m is a Lorentz invariant — it does not change with velocity. What grows with velocity is the effective inertia γm, because the relationship between force, acceleration, and velocity is fundamentally different in relativistic dynamics. This is sometimes described using 'relativistic mass' γm, but this is a pedagogical shortcut that can mislead. The physically meaningful invariant quantity is the rest mass m; γ captures the frame-dependent dynamical effect. The increased resistance to acceleration comes from the structure of relativistic momentum, not a literal change in mass.
Question 4 True / False
If you could apply infinite force to a massive object for finite time, you could in principle accelerate it to exactly the speed of light.
TTrue
FFalse
Answer: False
As v → c, γ → ∞, so p = γmv → ∞. Reaching c would require infinite momentum — and delivering infinite momentum requires infinite energy, regardless of the force magnitude or duration. Even an infinite force applied for finite time delivers finite impulse (momentum change), which cannot be infinite. The speed of light is an asymptote: you can always get closer, but each step requires more energy than the last, and the final step requires infinite energy. This is a structural consequence of relativistic momentum, not a practical engineering limit.
Question 5 Short Answer
Why can't a massive object be accelerated to the speed of light, even in principle? Frame your answer in terms of relativistic momentum.
Think about your answer, then reveal below.
Model answer: Relativistic momentum is p = γmv where γ = 1/√(1−v²/c²). As v approaches c, γ diverges: at 0.99c, γ ≈ 7; at 0.9999c, γ ≈ 71. The momentum p = γmv also diverges as v → c, meaning you would need infinite momentum — and thus infinite energy — to push an object to exactly c. Any finite applied force over any finite time delivers only finite momentum. The speed c is an asymptote that can be approached but never reached by a massive object. This is not a limitation of our technology but a structural consequence of how relativistic momentum grows with velocity.
This is also why massless particles (photons) always travel at exactly c — they are not being 'accelerated to c' but simply cannot travel at any other speed given their zero rest mass. The prohibition on massive objects reaching c and the requirement for massless objects to travel at c are two sides of the same relativistic coin.