Problem 4 - University Physics with Modern Physics

Question

A spaceship flies past Mars with a speed of 0.985 c 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\(\mu\) s. (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?

Step-by-Step Solution

Answer

(a) The observer on Mars measures the proper time. (b) The pilot measures 434.5 µs.

Step 1: Proper Time Identification

To determine who measures the proper time, we need to identify in which frame the events (signal light blinking on and off) occur at the same location. The observer on Mars sees the signal light blink on and then off at the same location on Mars' surface. Thus, the observer on Mars measures the proper time.

Step 2: Understanding Time Dilation

According to special relativity, time dilation occurs because time passes more slowly in a moving reference frame relative to a stationary observer. Thus, the time observed in the spaceship, which is moving at 0.985c, will be longer than the proper time measured by the observer on Mars.

Step 3: Apply the Time Dilation Formula

The time dilation formula is given by \( t = \frac{t_0}{\sqrt{1-v^2/c^2}} \), where \(t\) is the time measured by the observer in the moving frame (spaceship), \(t_0\) is the proper time (75.0 µs), and \(v\) is the speed of the spaceship (0.985 c).

Step 4: Calculation of Dilated Time

Substitute the values into the time dilation equation:\[ t = \frac{75.0 \mu s}{\sqrt{1-(0.985)^2}} \]Calculate \(1-(0.985)^2 = 1 - 0.970225 = 0.029775\).Then, calculate \(\sqrt{0.029775} \approx 0.172577\).Now, calculate \( t = \frac{75.0 \mu s}{0.172577} \approx 434.5 \mu s \).

Key Concepts

Special Relativity

Special Relativity is a fundamental theory in physics introduced by Albert Einstein in 1905. It changes the way we understand space and time. According to this theory, the laws of physics are the same for all observers, regardless of their relative motion. However, when an object moves at speeds close to the speed of light, significant differences arise. One of the most interesting effects of Special Relativity is time dilation. Time dilation means that a moving clock ticks more slowly compared to a stationary one. This helps us understand scenarios like the one described in the textbook exercise, where the speed of a spaceship influences the perception of time.

Proper Time

Proper Time is a concept in Special Relativity that identifies the time interval between two events as measured in the frame where the events occur at the same location. When discussing proper time, it's important to recognize that it is always the shortest possible time interval between events. In the context of the given exercise, the observer on Mars measures the proper time, as the light signal blinks on and off at the same spot on the Martian surface. The proper time is unaffected by any relativistic effects and provides a crucial reference point for understanding time measurements from other frames of reference.

Spaceship Motion

Spaceship Motion in the realm of Special Relativity is a fascinating subject. When a spaceship travels at a speed close to that of light, such as 0.985 times the speed of light (0.985c) in this exercise, it experiences significant relativistic effects. One of these effects, time dilation, becomes prominent. To correctly interpret physical phenomena, we need to consider how motion influences measurements like time and length. The speed of the spaceship means the observer inside it will measure time differently than an observer on Mars. This illustrates the impact of high-speed travel on time perception, building a practical example of Einstein's theories.

Relativistic Effects

Relativistic Effects refer to the phenomena that become noticeable when objects move at significant fractions of the speed of light. Some of the most notable effects include time dilation and length contraction. In the spaceship scenario, time dilation is crucial. While the Mars observer measures a 75.0 µs on-time for the light, the pilot on the spaceship perceives a longer duration due to the spaceship’s high velocity. Using the time dilation formula, \( t = \frac{t_0}{\sqrt{1-v^2/c^2}} \), with \( v = 0.985c \), allows us to calculate this increased time. Such calculations highlight the non-intuitive nature of time and space under relativistic conditions, emphasizing the need to consider these effects in high-speed contexts.