An observer in a coasting spacecraft moves toward a mirror...

An observer in a coasting spacecraft moves toward a mirror at speed v relative to the reference frame labeled by S in Figure P39.62. The mirror is stationary with respect to S. A light pulse emitted by the spacecraft travels toward the mirror and is reflected back to the spacecraft. The front of the spacecraft is a distance d from the mirror (as measured by observers in S) at the moment the light pulse leaves the spacecraft. What is the total travel time of the pulse as measured by observers in (a) the S frame and (b) the front of the spacecraft?

Key Concepts

Inertial Reference Frames

In special relativity, an inertial reference frame is a coordinate system in which an observer is either at rest or moving with a constant velocity, and where the fundamental laws of physics hold uniformly. In problems like the light pulse reflection scenario, analyzing events from different inertial frames (such as the S frame and the spacecraft frame) is crucial because measurements of time and distance can differ between them.

Invariance of the Speed of Light

One of the key postulates of special relativity states that the speed of light in a vacuum is constant (c) for all observers, regardless of their relative motion. This principle underpins the analysis of the light pulse's journey since both the observer in the spacecraft and observers in the S frame must use the same value for the speed of light when determining travel times.

Time Dilation

Time dilation is a relativistic effect where a clock moving relative to an observer ticks slower than a clock that is at rest in the observer’s frame. This concept is fundamental when comparing the time intervals measured by observers in different inertial frames, such as the proper time recorded in the moving spacecraft versus the dilated time observed in the stationary S frame.

Lorentz Transformation

The Lorentz transformation provides the mathematical framework to convert space and time coordinates of events from one inertial frame to another moving at a constant velocity relative to the first. These transformations help in understanding how the measured distances and time intervals associated with the light pulse event differ between observers in the spacecraft and the S frame.

Relative Motion and Signal Reflection

In this context, the analysis of the light pulse involves accounting for the relative motion between the source (the spacecraft) and the target (the mirror). The moving spacecraft and the stationary mirror result in different effective distances for the light to travel on its outbound and return trips. This concept is key in setting up and solving the equations for the light pulse’s round-trip journey time as seen in both frames.

Transcript

00:01 In this exercise, we have a spaceship represented in the figure by this green arrow that is traveling with the speed v relative to a reference frame s, as shown here in the picture.

00:14 And the spaceship is traveling towards a mirror that is abreast in the s reference frame.

00:20 When the distance between the spaceship and the mirror in the s reference frame is equal to d, the spaceship emits light towards the mirror.

00:33 That hits the mirror and then bounces back towards the spaceship.

00:38 And then the spaceship receives the beam of light again.

00:46 So in question a, we have to find how much time it takes for the light between the instant when the light is emitted by the spaceship and the instant when it's received back in the reference frame, where the mirror is still, that is in the s reference frame.

01:11 So basically, notice that the distance that the light travels in the s reference frame is equal to two times the distance, the natural distance between the spaceship and the mirror, minus the distance x that the spaceship travels between the emission of the light and the receiving of the light.

01:37 Notice that this distance x is equal to the speed of the spaceship times t.

01:45 So we have that d is equal to 2d minus v t.

01:59 Now also notice that the total time of travel, light t, is equal to the total distance that the light travels, which is d divided by the speed of light c.

02:12 Okay, so t is equal to 2d minus vt divided by c.

02:21 So i have t times c plus v is equal to 2d.

02:28 So t is equal to 2d divided by c plus v, which is the answer to question a.

02:41 In question b, we have to calculate how much time the same trajectory takes...

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