Question

An airplane whose rest length is \(42.4 \mathrm{~m}\) is moving with respect to the Earth at a constant speed of \(522 \mathrm{~m} / \mathrm{s}\). (a) By what fraction of its rest length will it appear to be shortened to an observer on Earth? (b) How long would it take by Earth clocks for the airplane's clock to fall behind by \(1 \mu \mathrm{s}\) ? (Assume that only special relativity applies.)

Short answer

For (a), the length of the airplane will appear to be shortened by \(0.000455\) or \(0.0455 \% \) of the length it is at rest to an observer on Earth. For (b), it would take approximately \(1.000001 \mu s\) by Earth clocks for the airplane's clock to fall behind by \(1 \mu s\).

Step-by-step solution

Identify Given Values

Identify the given values: Rest length of the plane \(L_0=42.4m\), Speed of the plane \(v=522m/s\), Time delay \(\Delta t=1 \mu s\). Also recall the speed of light \(c=3.0x10^{8}m/s\).

Calculate Length Contraction

Using the formula for the length contraction \(L=L_0/\gamma\) where \(\gamma=1/\sqrt{1-(v/c)^2}\), substitute the given values to find the length \(L\) and then calculate the percentage of rest length that it appears shortened, which is \((1 - L/L_0) x 100\% \)

Calculate Time Dilation

Using the formula for the time dilation \(\Delta T=\gamma*\Delta t\) (where \(\Delta T\) is the time according to an Earth observer and \(\Delta t\) is a proper time on the moving airplane, which is given), substitute the values to determine the time \(\Delta T\). Finally, find how much time the airplane's clock falls behind the earth clock, which is \(\Delta T - \Delta t\).

Key concepts

Understanding Length Contraction

Length contraction is a fascinating phenomenon predicted by Einstein's theory of special relativity. It occurs because objects moving close to the speed of light appear shorter in the direction of motion to a stationary observer.

• The formula involved in calculating the contracted length is: \[ L = \frac{L_0}{\gamma} \]
• Here, \( L_0 \) is the rest length (the length of the object in its own rest frame), and \( L \) is the length observed by someone stationary relative to the moving object.
• \( \gamma \) (gamma) is the Lorentz factor, which is defined as: \[ \gamma = \frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}} \] where \( v \) is the speed of the object and \( c \) is the speed of light.

The closer the object's speed is to the speed of light, the more significant the length contraction. It's crucial to note that this effect is not noticeable at everyday speeds; it becomes apparent only as speeds approach that of light.

In our exercise about an airplane with a rest length of 42.4 meters moving at 522 m/s, this contraction is very minute because the speed is quite low compared to the speed of light. This illustrates why relativistic effects like length contraction are only practically observable at speeds much closer to \( c \).

Exploring Time Dilation

Time dilation is another intriguing concept from special relativity. It describes how time passes differently for observers in different states of motion.

• The time dilation formula is: \[ \Delta T = \gamma \times \Delta t \]
• Here, \( \Delta T \) is the time interval as measured by a stationary observer (on Earth, in this case), and \( \Delta t \) is the proper time, which is the time interval measured by someone moving with the clock (on the airplane).

As with length contraction, the Lorentz factor \( \gamma \) affects the result, showing how much slower time appears to pass on the moving airplane from the perspective of an Earth observer.

Even at speeds as low as those in our example (522 m/s for the airplane), the concept of time dilation teaches us that the airplane’s clock runs slightly slower than the Earth's clock, though the effect is minor. This principle has profound implications; for instance, it explains why GPS satellites, moving rapidly relative to Earth, require correction for relativistic time effects to remain accurate.

Understanding Relativistic Effects

Relativistic effects encompass the differences in how space and time are perceived by observers in different inertial frames, especially when these frames are moving at high velocities relative to one another.

• The two core relativistic effects—length contraction and time dilation—are a direct consequence of the light-speed limit imposed by special relativity.
• These effects are intricately linked; one cannot occur without the other, preserving the consistency and integrity of the laws of physics across all frames of reference.

The reason these effects are not obvious in our daily lives is because they only become significant at speeds approaching the speed of light. However, they are crucial for technologies like particle accelerators and the Global Positioning System (GPS).

In the airplane example, although the speeds involved don't reach fractions anywhere near the speed of light, studying such scenarios provides clarity on how much reality differs at those higher velocities. By understanding these effects, one appreciates the universe's fascinating nature and the insights that special relativity brings to our comprehension of space and time.