Problem 65 - 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 $$ L(v)=10 \sqrt{1-\frac{v^{2}}{c^{2}}} $$ where \(c\) is the speed of light. $$ \begin{array}{l}{\text { (a) Find } L(0.5 c), L(0.75 c), \text { and } L(0.9 c)} \\ {\text { (b) How does the length of an object change as its velocity }} \\ {\text { increases? }}\end{array} $$

Step-by-Step Solution

Answer

Lengths: 8.66 m at 0.5c, 6.56 m at 0.75c, 4.36 m at 0.9c. Length decreases as velocity increases.

Step 1: Identify constants and variables

The function for length reduction due to relativity is given as \( L(v) = 10 \sqrt{1-\frac{v^{2}}{c^{2}}} \). Here, the length at rest \( L_0 = 10 \text{ m} \) and \( c \) is the speed of light. The variable \( v \) represents the velocity of the object as a fraction of the speed of light.

Step 2: Calculate L(0.5 c)

Substitute \( v = 0.5c \) into the function:\[ L(0.5c) = 10 \sqrt{1-\frac{(0.5c)^{2}}{c^{2}}} = 10 \sqrt{1-\frac{0.25c^{2}}{c^{2}}} = 10 \sqrt{0.75} = 10 \times \sqrt{0.75} \approx 8.66 \text{ m} \]

Step 3: Calculate L(0.75 c)

Substitute \( v = 0.75c \) into the function:\[ L(0.75c) = 10 \sqrt{1-\frac{(0.75c)^{2}}{c^{2}}} = 10 \sqrt{1-\frac{0.5625c^{2}}{c^{2}}} = 10 \sqrt{0.4375} = 10 \times \sqrt{0.4375} \approx 6.56 \text{ m} \]

Step 4: Calculate L(0.9 c)

Substitute \( v = 0.9c \) into the function:\[ L(0.9c) = 10 \sqrt{1-\frac{(0.9c)^{2}}{c^{2}}} = 10 \sqrt{1-\frac{0.81c^{2}}{c^{2}}} = 10 \sqrt{0.19} = 10 \times \sqrt{0.19} \approx 4.36 \text{ m} \]

Step 5: Analyze the effect of increasing velocity

As the velocity \( v \) of the object increases as a fraction of \( c \), the value within the square root \( 1-\frac{v^{2}}{c^{2}} \) decreases, causing the overall length \( L(v) \) to decrease. This demonstrates that the length contracts as the speed of the object approaches the speed of light.

Key Concepts

Length Contraction

Length contraction is a fascinating consequence of the Theory of Relativity, particularly at speeds approaching the speed of light. Imagine an object in space. When this object moves with speed comparable to light, its length appears shorter in the direction of motion to an outside observer. This isn't just an illusion—it reflects how space and time are interconnected.The mathematical expression for length contraction is:

• \[ L(v) = L_0 \sqrt{1 - \frac{v^2}{c^2}} \]

Here, \( L(v) \) is the contracted length at velocity \( v \), \( L_0 \) is the object's proper length (length at rest), and \( c \) is the speed of light (approximately \( 3 \times 10^8 \) meters per second).In our problem, as the speed of objects like spaceships reach fractions of the speed of light, the contracted length becomes measurable. This is important for understanding high-speed travel in physics and helps us realize how spacetime works together.

Speed of Light

The speed of light, symbolized as \( c \), is not just any speed—it's a universal constant that plays a crucial role in the laws governing the universe. Light travels at about 300,000 kilometers per second, a speed at which no material object can surpass. In relativity, \( c \) is the threshold for measuring all other speeds. When objects move at speeds comparable to light, their behavior deviates significantly from what we see at everyday, slower speeds.The speed of light impacts

• Length contraction: As speed approaches \( c \), lengths contract due to the formula \( L(v) = L_0 \sqrt{1 - \frac{v^2}{c^2}} \).
• Time dilation: Clocks on fast-moving objects tick slower compared to stationary ones.

These phenomena are the reasons we need the Theory of Relativity. They show how space and time are flexible under extreme conditions.

Velocity and Length Relationship

The relationship between velocity and length in the context of relativity is beautiful and strange. As an object's velocity increases close to the speed of light, its measured length shortens. This counterintuitive concept helps us appreciate the uniqueness of relativistic physics.When you substitute different fractions of \( c \) into the length contraction formula:

• At \( v = 0.5c \), there is a noticeable contraction.
• At \( v = 0.75c \), the contraction is more pronounced.
• At \( v = 0.9c \), the object appears significantly shortened.

Each increase in velocity enhances the effect of contraction proportionally, but never reaching zero length until \( v = c \), which physically can't occur.This relationship highlights that as velocity approaches \( c \), the entire perception of space transforms—a profound insight from relativity that challenges ordinary intuition and reshapes how we view movement through space.