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 approximately \( 3.23 \times 10^{-7} \) seconds later according to the pilot of cruiser B.

Step 1: Determine the Velocity of Cruiser A Relative to B

We are given a relative speed of cruiser A to cruiser B as \( v = 0.900c \). This is the speed at which cruiser A moves past cruiser B as observed from cruiser B.

Step 2: Understand Length Contraction

Because cruiser A is moving relative to cruiser B, cruiser A's length appears contracted from B's perspective. The contracted length \( L' \) is calculated using the formula \( L' = L \sqrt{1 - (v^2/c^2)} \), where \( L = 200 \text{ m} \) is the proper length and \( v = 0.900c \).

Step 3: Calculate the Contracted Length

Substitute the given values into the length contraction formula: \[ L' = 200 \sqrt{1 - (0.900)^2} = 200 \sqrt{1 - 0.81} = 200 \sqrt{0.19} \approx 200 \times 0.436 = 87.2 \text{ m} \]. Thus, the contracted length is approximately 87.2 m.

Step 4: Calculate the Time for Noses to Align Again

To find out how long it takes for the noses to align, calculate how much time it takes for cruiser A to cover the length of cruiser B as observed from cruiser B. That time \( t \) is given by \( t = \frac{L'}{v} \), where \( L' = 87.2 \text{ m} \) and \( v = 0.900c \).

Step 5: Substitute in the Values and Solve for Time

First, calculate the time: \[ t = \frac{87.2}{0.900 \times 3 \times 10^8} = \frac{87.2}{2.7 \times 10^8} \approx 3.23 \times 10^{-7} \text{ seconds} \]. Thus, the noses align approximately \( 3.23 \times 10^{-7} \) seconds later according to the pilot of cruiser B.

Key Concepts

Special relativity

Special relativity, a groundbreaking theory proposed by Albert Einstein, fundamentally changed our understanding of space and time. It introduces the idea that the laws of physics are the same for all non-accelerating observers. One of the key aspects of this theory is that the speed of light in a vacuum (denoted as \(c\)) is a universal constant at approximately \(3 \times 10^8\) meters per second.
This constant speed of light leads to several phenomena that might seem counterintuitive from a classical physics standpoint. One such phenomenon is time dilation, where time appears to slow down for an object in motion relative to an observer. Another is length contraction, where an object's length appears shorter when it is moving relative to an observer.

• Time and length are not absolutes but are relative to the observer’s frame of reference.
• Physics laws are consistent across all inertial frames of reference.

Special relativity fundamentally alters how we understand the interactions between space and time, providing a foundation for many advancements in modern physics.

Proper length

Proper length, often denoted as \(L\), is the length of an object measured by an observer at rest relative to the object. In the context of special relativity, an object's proper length is the longest length that can be measured, and it applies to situations where the observer and the object are both stationary with respect to each other.
When the object moves at high speeds, particularly close to that of light, observers who see the object in motion observe a phenomenon known as length contraction. The length of the object, from these observers' perspectives, appears shorter than the proper length.

• Proper length is the true length of the object from its own rest frame.
• Proper length is different from the contracted length observed during relative motion.

The formula for length contraction is \(L' = L \sqrt{1 - (v^2/c^2)}\), where \(v\) is the velocity of the object and \(c\) is the speed of light. This equation highlights how velocity affects the measured length in different frames of reference.

Relative velocity

Relative velocity is a measure of how fast one object is moving in relation to another. It becomes particularly important in the realm of special relativity where objects move at significant fractions of the speed of light.
In our exercise, the relative velocity of cruiser A as observed from cruiser B is given as \(v = 0.900c\). This means cruiser A is moving past cruiser B at 90% of the speed of light. Due to the high speed, relativistic effects like length contraction become noticeable.

• Relative velocity is the velocity of an object as observed from another moving reference frame.
• It plays a crucial role in relativistic physics, affecting both time and space measurements.

The concept underscores that velocities are not absolute but depend on the frame from which observations are made. This interrelation shapes our understanding of high-speed interactions and lies at the heart of many relativity-based calculations.