An observer in reference frame S notes that two events are separated in space by 180 m and in time by 0.80μs. How fast must reference frame S' be moving relative to S in order for an observer in S' to detect the two events as occurring at the same location in space?

Verified step by step guidance

1. Understand the problem: The observer in reference frame S' must see the two events occurring at the same location in space. This means the spatial separation between the events in S' (Δx') must be zero. Use the Lorentz transformation equations to relate the space and time coordinates of the events in S and S'.
2. Write the Lorentz transformation equation for spatial separation: Δx' = γ(Δx - vΔt), where Δx is the spatial separation in S (180 m), Δt is the time separation in S (0.80 μs = 0.80 × 10⁻⁶ s), v is the relative velocity between S and S', and γ = 1 / √(1 - v²/c²) is the Lorentz factor.
3. Set Δx' = 0 because the events must occur at the same location in S'. This simplifies the equation to 0 = γ(Δx - vΔt). Solve for v by isolating it: Δx = vΔt.
4. Substitute the known values for Δx and Δt into the equation: 180 m = v × (0.80 × 10⁻⁶ s). Rearrange to solve for v: v = Δx / Δt.
5. To ensure the result is consistent with relativistic principles, verify that the calculated v is less than the speed of light c (approximately 3.00 × 10⁸ m/s). If v < c, the solution is valid, and this is the required relative velocity for S' to observe the events at the same location.

Key Concepts

Relativity of Simultaneity

The relativity of simultaneity is a fundamental concept in Einstein's theory of special relativity, which states that events that are simultaneous in one reference frame may not be simultaneous in another. This occurs because the speed of light is constant for all observers, leading to differences in the perception of time and space depending on the observer's relative motion.

Lorentz Transformation

The Lorentz transformation equations relate the space and time coordinates of events as observed in different inertial frames moving at a constant velocity relative to each other. These equations account for the effects of time dilation and length contraction, allowing for the calculation of how measurements of time and distance change between observers in relative motion.

Velocity Addition in Relativity

In special relativity, the classical notion of adding velocities does not hold true at high speeds. Instead, the relativistic velocity addition formula must be used, which ensures that the resultant velocity never exceeds the speed of light. This concept is crucial for understanding how the relative motion of different reference frames affects the observation of events.