Question

If a muon is moving at \(90.0 \%\) of the speed of light, how does its measured lifetime compare to when it is in the rest frame of a laboratory, where its lifetime is \(2.2 \cdot 10^{-6}\) s?

Short answer

Answer: When moving at 90% the speed of light, the measured lifetime of a muon is approximately 5.046 microseconds, which is more than 2 times longer than its lifetime in the rest frame (2.2 microseconds).

Step-by-step solution

Identify the time dilation formula

The time dilation formula, as per special relativity, is given by: \(T = T_{0} \cdot \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\), where T is the dilated time (measured time), \(T_{0}\) is the proper time (time in the rest frame), v is the relative velocity of the moving frame, and c is the speed of light.

Determine the moving frame's relative velocity

The muon is moving at 90% of the speed of light. To find its relative velocity v, we multiply the speed of light (c) by 0.9 (90% in decimal form): \(v = 0.9c\).

Substitute values in the time dilation formula

We have the proper time \(T_{0} = 2.2 \cdot 10^{-6}\) s, and the relative velocity \(v = 0.9c\). Substitute these values into the equation: \(T = (2.2 \cdot 10^{-6}) \cdot \frac{1}{\sqrt{1-\frac{(0.9c)^2}{c^2}}}\).

Simplify the equation

Now we need to simplify the expression in the equation: \(T = (2.2 \cdot 10^{-6}) \cdot \frac{1}{\sqrt{1-\frac{(0.81c^2)}{c^2}}}\).

Cancel the terms and calculate the time dilation factor

The \(c^2\) in the numerator and the denominator cancel each other out: \(T = (2.2 \cdot 10^{-6}) \cdot \frac{1}{\sqrt{1-0.81}}\). Calculate the time dilation factor: \(\frac{1}{\sqrt{1-0.81}} = \frac{1}{\sqrt{0.19}} \approx 2.294\).

Determine the measured lifetime of the moving muon

Multiply the proper time by the time dilation factor: \(T \approx (2.2 \cdot 10^{-6}) \cdot 2.294 \approx 5.046 \cdot 10^{-6}\) s. So, the measured lifetime of the muon when it is moving at 90% the speed of light is approximately 5.046 microseconds.

Compare the measured lifetime to the lifetime in the rest frame

The lifetime in the rest frame of the laboratory is 2.2 microseconds, while the measured lifetime when it is moving at 90% of the speed of light is approximately 5.046 microseconds. This means that the muon's lifetime appears to be more than 2 times longer when moving at a high speed compared to its lifetime in the rest frame.

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics, first proposed by Albert Einstein in 1905. It revolutionized the way we understand space, time, and motion. At its core, special relativity reveals how the laws of physics are the same for all observers, regardless of their relative velocity. One of the crucial postulates is that the speed of light is constant for all observers, no matter how fast they move relative to each other. This leads to some fascinating outcomes, including time dilation.

Time dilation is the concept that time can pass at different rates for different observers, depending on their relative velocity. When an object moves close to the speed of light, time appears to pass more slowly for it compared to a stationary observer. This effect has been confirmed by many experiments, making it a key prediction of special relativity. Understanding special relativity allows us to explore peculiar phenomena such as how fast-moving particles behave and is critical for interpreting results in high-energy physics and cosmology. By diving deep into special relativity, we gain insight into the relationship between time, space, and motion.

Muon Lifetime

Muons are elementary particles similar to electrons, but they are around 200 times heavier, and they have a short lifespan. In the context of special relativity, muons provide compelling evidence for time dilation through their observed lifetimes. Normally, muons have a lifetime of about 2.2 microseconds when at rest. However, when they move at speeds close to that of light, such as 90% of the speed of light, their lifetimes as observed from a stationary reference frame—like a lab—increase significantly.

This change is a direct result of time dilation as predicted by special relativity. For example, as demonstrated in the solution to the exercise, a muon moving at 90% the speed of light will have a measured lifetime of about 5.046 microseconds in the laboratory rest frame. This is because time effectively 'stretches' for the muon when it moves at such high speeds. The study of muons and their behavior in motion not only supports the theory of special relativity but also helps physicists probe fundamental questions about the universe and the nature of particles.

Speed of Light

The speed of light, denoted by the symbol \(c\), is one of the most important constants in the universe. It is approximately equal to \(299,792,458\) meters per second. This constant represents the ultimate speed limit in the universe, according to the theory of special relativity. No information or matter can travel faster than the speed of light. This limitation has profound implications for our understanding of space, time, and the universe at large.

The constancy of the speed of light ensures that it remains the same for all observers, regardless of their motion or the motion of the light source. This invariance is a cornerstone of special relativity and leads to phenomena like time dilation and length contraction, allowing us to better comprehend the universe at both very high speeds and in strong gravitational fields. Light's speed defines the maximum rate at which information can be transmitted, affecting fields as varied as telecommunications, astrophysics, and space exploration. Understanding the speed of light helps us grasp not only practical applications but also the philosophical implications of our place in the space-time continuum.