A firecracker explodes in reference frame S at t = 1.0 s. A second firecracker explodes at the same position at t = 3.0 s. In reference frame S', which moves in the x-direction at speed v, the first explosion is detected at x' = 4.0 m and the second at x' = -4.0 m. What is the speed of frame S' relative to frame S?

Verified step by step guidance

1. Start by identifying the key concepts involved in the problem. This is a relativistic problem involving the Lorentz transformation equations, which relate the coordinates (time and position) in one inertial reference frame to those in another moving at a constant velocity relative to the first.
2. Write down the Lorentz transformation equations for position and time. For the x-coordinate and time, they are: and , 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.
3. Use the given data to set up equations. In frame S, the first explosion occurs at (x = 0, t = 1.0 s) and the second explosion occurs at (x = 0, t = 3.0 s). In frame S', the first explosion is detected at (x' = 4.0 m) and the second at (x' = -4.0 m). Substitute these values into the Lorentz transformation equations for x' and solve for v.
4. For the first explosion, substitute x = 0, t = 1.0 s, and x' = 4.0 m into the equation . This simplifies to <4.0 = γ(-v × 1.0)>. Similarly, for the second explosion, substitute x = 0, t = 3.0 s, and x' = -4.0 m into the same equation, which simplifies to <-4.0 = γ(-v × 3.0)>. These two equations can be solved simultaneously to find v.
5. Divide the two equations to eliminate γ and solve for v. This will give you the relative speed of frame S' with respect to frame S. Remember to check that the value of v is less than the speed of light, c, as required by special relativity.

Key Concepts

Reference Frames

A reference frame is a perspective from which measurements are made, including time and position. In physics, different reference frames can yield different observations of events, especially when they are in relative motion. Understanding how events are perceived in different frames is crucial for solving problems involving motion and relativity.

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, which are essential for understanding how measurements of time and distance change between moving observers.

Velocity Addition

Velocity addition is the process of determining the resultant velocity of an object when observed from different reference frames. In special relativity, the classical addition of velocities does not hold; instead, the relativistic velocity addition formula must be used to accurately calculate the speed of one frame relative to another, especially when approaching the speed of light.