Galilean and special relativity practice questions.

A spacecraft moves away from Earth at 0.60c. It launches a probe in its direction of motion at 0.60c relative to the spacecraft. Determine the speed of the probe as measured from Earth.

• 1.2c
• 0.60c
• 0.88c
• 0.75c

Answer: 0.88c

Relativistic velocity addition gives u = (0.60c + 0.60c)/(1 + 0.60 × 0.60) = 1.2c/1.36 ≈ 0.88c. Galilean addition (1.2c) fails because no object can exceed c.

The proper mean lifetime of a muon is 2.2 μs. A muon travels through the atmosphere at 0.98c, for which the Lorentz factor is 5.0. The mean lifetime of this muon measured by an observer on the ground is approximately:

• 0.44 μs
• 2.2 μs
• 4.4 μs
• 11 μs

Answer: 11 μs

Time dilation: Δt = γΔt₀ = 5.0 × 2.2 μs = 11 μs. Dividing instead of multiplying (0.44 μs) is the classic error — the moving muon's clock runs slow, so the ground observer measures a longer lifetime.

A rod has a proper length of 5.0 m. It moves parallel to its length at 0.80c relative to an observer. Calculate the length of the rod measured by this observer.

• 3.0 m
• 4.0 m
• 5.0 m
• 8.3 m

Answer: 3.0 m

At 0.80c the Lorentz factor is γ = 1/√(1 − 0.64) = 1/0.60. Length contraction gives L = L₀/γ = 5.0 × 0.60 = 3.0 m; multiplying by γ (8.3 m) is a common sign-of-effect error.

Which of the following is one of the two postulates of special relativity?

• The speed of light in vacuum is the same for all observers, including accelerating ones
• Time intervals between events are absolute and identical for all observers
• The speed of light in vacuum is the same for all inertial observers
• The laws of physics take their simplest form in the reference frame of the Earth

Answer: The speed of light in vacuum is the same for all inertial observers

The postulates are: (1) the laws of physics are identical in all inertial frames, and (2) the speed of light in vacuum is the same for all inertial observers. Special relativity applies to inertial frames only, and absolute time is precisely what it abandons.

In frame S, two lamps at different positions flash at exactly the same time. Frame S' moves at constant velocity relative to S along the line joining the lamps. Which statement is correct for an observer in S'?

• The flashes are simultaneous in S' as well, because simultaneity is absolute
• The flashes are not simultaneous in S'
• The flashes are simultaneous in S' only if the observer is midway between the lamps
• The order of the flashes in S' is always the same as the order of the lamp positions in S

Answer: The flashes are not simultaneous in S'

The relativity of simultaneity: events at different positions that are simultaneous in one inertial frame are not simultaneous in a frame moving relative to it (Δt' = γ(Δt − vΔx/c²) ≠ 0 when Δt = 0 but Δx ≠ 0).

Two events are separated by time interval Δt and spatial separation Δx as measured in some inertial frame. Which quantity has the same value in every inertial reference frame?

• (cΔt)² − (Δx)²
• (cΔt)² + (Δx)²
• Δt
• Δx

Answer: (cΔt)² − (Δx)²

The spacetime interval (cΔt)² − (Δx)² is invariant under the Lorentz transformation. Δt and Δx individually change from frame to frame, and the sum of squares (a Euclidean distance) is not invariant.