Question

A spaceship flies past Mars with a speed of 0.985c relative to the surface of the planet. When the spaceship is directly overhead, a signal light on the Martian surface blinks on and then off. An observer on Mars measures that the signal light was on for 75.0 ms. (a) Does the observer on Mars or the pilot on the spaceship measure the proper time? (b) What is the duration of the light pulse measured by the pilot of the spaceship?

Short answer

(a) Mars observer measures proper time; (b) Duration for pilot is 432 ms.

Step-by-step solution

Understanding Proper Time

Proper time, in the context of special relativity, is the time interval measured by an observer at rest relative to the event taking place. Here, the event is the blinking of the signal light on Mars. Since the observer on Mars is at rest relative to the light, the 75.0 ms measured by this observer is the proper time.

Identifying Reference Frames

There are two reference frames: the Martian observer's frame (where the light blinks on Mars) and the spaceship pilot's frame (where the spaceship is moving at 0.985c). We know that observers in the moving frame perceive the duration as dilated compared to the proper time.

Applying Time Dilation Formula

The time dilation formula is given by \[T = \frac{T_0}{\sqrt{1 - \frac{v^2}{c^2}}}\]where \(T_0\) is the proper time, \(v\) is the velocity of the spaceship, and \(c\) is the speed of light. Here, \(T_0 = 75.0\) ms and \(v = 0.985c\).

Calculating the Dilated Time

Substitute the known values into the formula:\[T = \frac{75.0\, \text{ms}}{\sqrt{1 - (0.985)^2}}\]Calculating this gives:\[T \approx \frac{75.0\, \text{ms}}{\sqrt{1 - 0.970225}}\]\[T \approx \frac{75.0\, \text{ms}}{0.1736}\]\[T \approx 432\, \text{ms}\]Thus, the duration measured by the pilot is approximately 432 milliseconds.

Key concepts

Special Relativity

Special relativity is a scientific theory formulated by Albert Einstein in 1905. This theory revolutionized our understanding of time, space, and how objects move through them. One of the key ideas of special relativity is that the laws of physics are the same for all observers, regardless of their motion, as long as they are not accelerating. This means that the speed of light, denoted as \( c \), remains constant for all observers, regardless of their own speed or direction.
One of the results of special relativity is the phenomenon of time dilation, which explains how an observer in motion relative to a stationary observer will measure time differently. In simpler terms, a clock moving at high speeds will tick slower compared to a clock that is stationary, when observed from the stationary reference frame. This concept challenges our intuitive understanding of time and underscores the interconnectedness of space and time.
Understanding this theory can help explain why the spaceship pilot and the observer on Mars measure different times for the same event, such as the blinking of a light on Mars. The differences arise because of their distinct motions relative to each other. Thus, special relativity provides the framework to understand time dilation and how it applies to different reference frames.

Proper Time

In the realm of special relativity, proper time is defined as the time interval between two events as measured by an observer who is at rest relative to the events occurring at the same spatial location. To grasp this concept, imagine a clock that is stationary with respect to the observer; the time elapsed on this clock is the proper time.
In the problem concerning the spaceship and Mars, the signal light's blinking is considered an event. The observer standing on Mars, who sees the signal light blink on and off without moving relative to the light source, measures the proper time, which is given as 75.0 milliseconds (ms).
Proper time helps us distinguish between various measurements made by observers situated in different reference frames. Only the observer for whom the events occur at the same location in space measures the proper time. This measurement acts as a baseline for comparing other observations from moving reference frames, like that of the spaceship pilot.

Reference Frames

Reference frames are crucial concepts in physics, especially in understanding motion and relativity. A reference frame is essentially a viewpoint, a coordinate system relative to which an observer measures positions, speeds, and other physical quantities. In special relativity, these frames are particularly vital as they dictate how events are perceived by observers.
In our spaceship problem, we have two primary reference frames: one attached to the Martian observer and the other to the spaceship pilot. The Martian observer's frame is stationary relative to the blinking light, making it the rest frame for the event. In contrast, the spaceship pilot's frame is moving at 0.985c, which is 98.5% the speed of light, making it a moving frame relative to the Martian observer.
Different reference frames explain why the time measured by the pilot (432 ms) differs from the proper time measured on Mars (75.0 ms). The time dilation effect occurs because the pilot's frame is moving at a substantial fraction of the speed of light, leading to a longer duration of the event measured. Thus, understanding reference frames allows us to accurately predict and explain how observations change due to relative motion.