Problem 90 - Fundamentals of Physics

Question

Space cruisers \(A\) and \(B\) are moving parallel to the positive direction of an \(x\) axis. Cruiser \(A\) is faster, with a relative speed of \(v=0.900 c\), and has a proper length of \(L=200 \mathrm{~m}\). According to the pilot of \(A\), at the instant \((t=0)\) the tails of the cruisers are aligned, the noses are also. According to the pilot of \(B\), how much later are the noses aligned?

Step-by-Step Solution

Answer

The noses align in about 320 nanoseconds from pilot B's perspective.

Step 1: Identify Variables and Concepts

Firstly, identify the known values and the concepts involved. You are given that the relative speed between the two cruisers is \(v = 0.900c\), where \(c\) is the speed of light, and the proper length of cruiser \(A\) is \(L = 200 \, \text{m}\). The problem involves time dilation and length contraction due to the relativistic speeds of the cruisers.

Step 2: Apply Length Contraction

Since cruiser \(A\) is observed by pilot \(B\), from pilot \(B\)'s perspective, the length of cruiser \(A\) is contracted. Calculate the contracted length using the length contraction formula: \(L' = L \sqrt{1 - \frac{v^2}{c^2}}\). Substitute in the values: \[ L' = 200 \, \text{m} \times \sqrt{1 - (0.900)^2} = 200 \, \text{m} \times \sqrt{1 - 0.81} = 200 \, \text{m} \times \sqrt{0.19} \approx 86.4 \, \text{m}. \]

Step 3: Calculate Time Interval

Since the contracted length of cruiser \(A\) is \(86.4 \, \text{m}\), this is the distance that needs to be covered for the noses to align from pilot \(B\)'s perspective. Use the formula \(\Delta t = \frac{L'}{v}\). Calculate the time \(\Delta t\) it takes for cruiser \(A\) to travel this distance relative to cruiser \(B\): \[ \Delta t = \frac{86.4 \, \text{m}}{0.900c} = \frac{86.4}{0.900 \times 3 \times 10^8} \, \text{s} \approx 3.2 \times 10^{-7} \, \text{s}. \]

Step 4: Give the Final Answer

Finally, express the time interval \(\Delta t\) for clarity, which is the time it takes for the noses of the cruisers to become aligned from pilot \(B\)'s perspective. This corresponds to approximately \(320 \, \text{nanoseconds}\), confirming the derivation and calculation.

Key Concepts

Length Contraction

Length contraction is a fascinating phenomenon that occurs at high speeds, particularly close to the speed of light. When an object moves at relativistic speeds, its length along the direction of motion appears shortened from the perspective of a stationary observer. This doesn't mean the object physically shrinks. Instead, it's a part of Einstein's theory of relativity. To find the contracted length, we use the formula:\[ L' = L \sqrt{1 - \frac{v^2}{c^2}} \]

• Here,
  • L is the proper length (the length of the object in its rest frame),
  • v is the velocity of the object,
  • c is the speed of light, and
  • L' is the observed contracted length.

When applying this to spaceship A from cruiser B's point of view, for instance, lengths appear less than their proper lengths due to the high speed (0.900c), resulting in a contracted length of approximately 86.4 m. This contraction only affects the directions along the direction of movement.

Relativistic Speeds

Relativistic speeds refer to velocities that are a significant fraction of the speed of light. At such speeds, classical mechanics fail to describe motion accurately, and relativistic mechanics, part of Einstein's special theory of relativity, must be applied. The effects that come into play are not only length contraction but also time dilation and mass increase, which may not be intuitive.
Relativistic effects become significant as an object's speed approaches the speed of light (denoted by \(c\), approximately \(3 \times 10^8 \, \text{m/s}\)). For cruiser A moving at 0.900c, relativistic effects are very pronounced, altering both the perceptions of space and time for observers in different reference frames. For example, from cruiser B's point of view, cruiser A not only appears shorter in length but also operates within a different time frame due to these speeds.

Proper Length

The concept of proper length is central in understanding relativistic effects like length contraction. Proper length is the length of an object measured by an observer at rest relative to the object. In other words, it's the "true" length of the object when it's not moving in the observer's frame of reference.
For cruiser A, its proper length is 200 meters. This length is measured when the cruiser is at rest with respect to the observer. However, once cruiser A starts moving near light speed, observers in different frames, such as cruiser B's pilot, measure a shorter length for cruiser A due to relativistic effects.

Relativity

Relativity, fundamentally, is the framework for understanding how space and time are linked for objects moving at a consistent speed in a straight line. Einstein's special theory of relativity dramatically shifts our intuitive understanding of motion and introduces the revolutionary idea that the laws of physics are the same for all observers.

• Key tenets of relativity include:
  • The speed of light is constant for all observers, regardless of their motion relative to the light source.
  • These principles lead to effects like time dilation (moving clocks tick slower) and length contraction (as covered).

Special relativity ensures that all physical quantities such as length, mass, and time can differ based on the frame of the observer, but the laws themselves remain consistent. It's through this lens that we understand the alignment of cruiser noses, with cruiser A's observed behaviors being a direct application of these principles.