Rocket A moves at 0.9c to the left and rocket B moves at 0.9c to the right, as measured from Earth. How fast does an observer in rocket A measure rocket B to be moving?
A1.8c — the velocities add directly since we are combining speeds in a single calculation
B0.9c — velocities cannot add at all in special relativity
CApproximately 0.994c — the relativistic addition formula applies and the denominator pulls the result below c
DExactly c — any two velocities near c combine to exactly c
The Galilean sum 0.9c + 0.9c = 1.8c describes only the rate at which the separation between the rockets grows as measured from Earth — a valid coordinate quantity in that frame, but not a velocity of either rocket. To find rocket B's speed in rocket A's rest frame, apply the relativistic formula: (0.9 + 0.9)/(1 + 0.9×0.9) = 1.8/1.81 ≈ 0.994c. The denominator (1 + uv/c²) is what pulls the result back below c. This is not a coincidence — it is built into the Lorentz transformation structure.
Question 2 Multiple Choice
A spaceship traveling at 0.8c relative to Earth fires a laser beam forward. What speed does Earth measure the laser beam traveling?
A1.8c — the ship's speed adds directly to the speed of light
Bc — the speed of light is invariant in all inertial frames, and the formula confirms this
C0.2c — the beam's speed is reduced because the ship is moving toward the beam
DSlightly greater than c, but unmeasurable with current instruments
Apply the formula with u = c (speed of light in the ship's frame) and v = 0.8c (ship's speed): u′ = (c + 0.8c)/(1 + 0.8c·c/c²) = 1.8c/1.8 = c. The result is exactly c. This is not an independent postulate bolted onto the formula — it falls out naturally from the Lorentz transformation algebra. The denominator (1 + uv/c²) cancels the numerator's excess in exactly the right way whenever u = c.
Question 3 True / False
The rate at which the separation between two rockets increases, as measured by a stationary observer, can exceed c even though neither rocket moves faster than c.
TTrue
FFalse
Answer: True
In a single inertial frame, the rate of change of separation between two objects is a coordinate quantity, not the speed of any physical thing. If rocket A moves left at 0.9c and rocket B moves right at 0.9c as measured from Earth, Earth observes their separation growing at 1.8c per unit time. No object is moving at 1.8c — this is just arithmetic on two separate velocities in the same frame. Special relativity restricts the speed of a physical object in any inertial frame to less than c; it does not restrict coordinate separation rates, which is why closing speed and relative velocity must be carefully distinguished.
Question 4 True / False
If object A moves at 0.6c relative to Earth and object B moves at 0.8c in the same direction relative to Earth, then object B moves at 0.2c relative to object A.
TTrue
FFalse
Answer: False
The naive subtraction 0.8c − 0.6c = 0.2c is the Galilean result. Applying the relativistic formula: (0.8c − 0.6c)/(1 − 0.6×0.8) = 0.2c/(1 − 0.48) = 0.2c/0.52 ≈ 0.385c. The relativistic result is nearly twice the Galilean estimate here, illustrating that the correction is non-trivial even at moderate speeds. The denominator (1 − uv/c²) < 1 when both objects move in the same direction, which amplifies the relative velocity above the naive difference — though still keeping it below c.
Question 5 Short Answer
Why does applying the relativistic velocity addition formula with u = c always yield c, regardless of the value of v?
Think about your answer, then reveal below.
Model answer: Substituting u = c into the formula gives (c + v)/(1 + vc/c²) = (c + v)/(1 + v/c) = c(1 + v/c)/(1 + v/c) = c. The factor (1 + v/c) appears identically in both numerator and denominator and cancels exactly. This is not a coincidence — the Lorentz transformation from which the formula is derived was constructed precisely to preserve the invariance of c. The cancellation is a mathematical expression of the second postulate of special relativity: light has the same speed c in every inertial frame.
This result shows that the relativistic addition formula does not just approximate c-invariance or treat it as a boundary condition — it builds c-invariance into its algebraic structure. The formula reduces to Galilean addition at low speeds (when uv/c² ≪ 1) and enforces c-invariance at the extreme (when u = c). It is consistent both with everyday experience and with the foundational postulate of special relativity.