Problem 75 - Algebra and Trigonometry

Question

Relativity According to the Theory of Relativity, the length \(L\) of an object is a function of its velocity \(v\) with respect to an observer. For an object whose length at rest is \(10 \mathrm{m},\) the function is given by $$L(v)=10 \sqrt{1-\frac{v^{2}}{c^{2}}}$$ $$L(v)=10 \sqrt{1-\frac{v^{2}}{c^{2}}}$$ where \(c\) is the speed of light \((300,000 \mathrm{km} / \mathrm{s}) .\)(b) How does the length of an object change as its velocity increases?

Step-by-Step Solution

Answer

The length of the object contracts as its velocity increases, approaching zero as velocity approaches the speed of light.

Step 1: Understand the Function

The function given is \( L(v) = 10 \sqrt{1 - \frac{v^2}{c^2}} \), where \( L(v) \) is the length at velocity \( v \) and \( L(0) = 10 \) m when the object is at rest. This represents the relativistic effect of length contraction.

Step 2: Concept of Length Contraction

According to the theory, the length of an object contracts (appears shorter) when observed at velocities closer to the speed of light \( c \). As \( v \) increases, \( 1 - \frac{v^2}{c^2} \) decreases, resulting in a smaller value under the square root.

Step 3: Analyze the Denominator

The term \( \frac{v^2}{c^2} \) increases as velocity \( v \) increases, meaning \( \sqrt{1 - \frac{v^2}{c^2}} \) decreases because the fraction becomes larger, thus decreasing \( L(v) \).

Step 4: Limit Analysis

Consider \( v \) approaching \( c \): \( \frac{v^2}{c^2} \rightarrow 1 \), making \( \sqrt{1 - \frac{v^2}{c^2}} \rightarrow 0 \). Thus, \( L(v) \) approaches 0. When \( v \) is much less than \( c \), \( \frac{v^2}{c^2} \approx 0 \), and \( L(v) \) approximately equals 10 m.

Key Concepts

Length Contraction

One fascinating aspect of Einstein's Theory of Relativity is the concept of length contraction. Imagine observing a spaceship zooming through space at a speed close to the speed of light. To you, as a stationary observer, the spaceship would appear shorter along its direction of motion than it would if it were at rest.
This phenomenon occurs because as the spaceship's velocity \( v \) increases, the factor \( 1 - \frac{v^2}{c^2} \) decreases. Therefore, the length \( L(v) = 10 \sqrt{1 - \frac{v^2}{c^2}} \) becomes smaller.

• When \( v = 0 \), the spaceship remains its original length of 10 meters.
• As \( v \) increases, the observed length \( L(v) \) shrinks.
• When \( v \) approaches the speed of light \( c \), \( L(v) \) approaches zero.

This illustrates how objects traveling at high speeds experience a contraction in length from the viewpoint of a stationary observer, aligning with the predictions of relativistic physics.

Relativistic Effects

Relativistic effects are phenomena that occur when objects travel at speeds close to that of light. These effects fundamentally change how we perceive time and space, significantly differing from classical Newtonian mechanics.
One of the key ideas involves not just length contraction, but also time dilation and mass increase. These all arise because the laws of physics and the speed of light remain invariant, no matter the observer's frame of reference.

• As speed increases, time for the moving object slows down relative to a stationary observer (time dilation).
• The mass of the moving object increases, requiring more energy to continue accelerating.
• Length contraction, as previously mentioned, causes the object to appear shorter in the direction of motion.

These combined effects create a fascinating picture of the universe at relativistic speeds, challenging our everyday expectations and leading to groundbreaking discoveries. Relativity has reshaped our understanding of how the universe operates at its most fundamental levels.

Velocity and Speed of Light

A core principle of the Theory of Relativity is that no object with mass can travel at or faster than the speed of light, \( c = 300,000 \text{ km/s} \). This speed limit is fundamental and dictates many relativistic effects.
Consider the spaceship from our earlier example. As its velocity \( v \) approaches the speed of light, its relativistic mass increases dramatically, requiring more and more energy to continue accelerating.

• The equation \( L(v) = 10 \sqrt{1 - \frac{v^2}{c^2}} \)
illustrates that as \( v \rightarrow c \), \( \frac{v^2}{c^2} \rightarrow 1 \), thus making the multiplier 0 and shrinking the object’s length to nearly zero.
• Any effort to reach the speed of light is met with resistance due to increasing relativistic mass, making \( c \) unreachable for objects with mass.

The restriction of velocity reinforces the idea that the speed of light is an ultimate cosmic speed limit, explaining why light behaves the same for all observers, regardless of their relative motion. This understanding is crucial for grasping numerous modern physics concepts, including the structure of the universe, particle physics, and cosmology.