Ch 36: Special Relativity

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 36: Special Relativity

Problem 45b

Chapter 36, Problem 45b

Two events in reference frame S occur 10 μs apart at the same point in space. The distance between the two events is 2400 m in reference frame S'. What is the velocity of S' relative to S?

Verified step by step guidance

Step 1: Identify the key concepts involved in the problem. This is a relativistic problem involving time dilation and length contraction. The relationship between the two reference frames S and S' is governed by the Lorentz transformation equations.

Step 2: Write down the given information. In reference frame S, the time interval between the two events is Δt = 10 μs (10 × 10⁻⁶ s), and the spatial separation is Δx = 0 m (since the events occur at the same point in space). In reference frame S', the spatial separation is Δx' = 2400 m.

Step 3: Use the Lorentz transformation equation for spatial separation: Δx' = γ(Δx - vΔt), where γ = 1 / √(1 - v²/c²) is the Lorentz factor, v is the relative velocity between the frames, and c is the speed of light. Substitute Δx = 0 into the equation to simplify it to Δx' = γ(-vΔt).

Step 4: Rearrange the equation to solve for v. Start by expressing γ in terms of Δx' and Δt: γ = Δx' / (-vΔt). Then use the definition of γ (γ = 1 / √(1 - v²/c²)) to eliminate γ and solve for v.

Step 5: Substitute the known values (Δx' = 2400 m, Δt = 10 × 10⁻⁶ s, and c = 3 × 10⁸ m/s) into the equation derived in Step 4. Perform algebraic manipulations to isolate v and determine the relative velocity of S' with respect to S.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Lorentz Transformation

The Lorentz transformation equations relate the space and time coordinates of events as observed in different inertial frames moving relative to each other at a constant velocity. They account for the effects of time dilation and length contraction, which are fundamental in the theory of special relativity. These transformations are essential for understanding how measurements of time and distance change when switching between frames.

Time Dilation

Time dilation is a phenomenon predicted by Einstein's theory of special relativity, where time is observed to pass at different rates in different inertial frames. Specifically, a clock moving relative to an observer will appear to tick slower than a stationary clock. This concept is crucial for analyzing events that occur at different times in different reference frames, as it affects the perceived intervals between events.

Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. In the context of special relativity, it is important to calculate how fast one reference frame is moving relative to another, especially when considering the effects of time and space. Understanding relative velocity helps in determining how events are perceived in different frames, which is key to solving problems involving multiple reference frames.

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