Problem 3 - University Physics with Modern Physics

Question

How fast must a rocket travel relative to the earth so that time in the rocket "slows down" to half its rate as measured by earth-based observers? Do present-day jet planes approach such speeds?

Step-by-Step Solution

Answer

A rocket must travel at approximately \(2.598 \times 10^8\) m/s, much faster than any jet plane.

Step 1: Understanding Time Dilation

The phenomenon we're dealing with is called time dilation, which is a part of Einstein's theory of relativity. Time dilation occurs when there's a relative velocity between two observers, and it's described by the equation: \[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \]where \( t' \) is the time observed in the moving frame (rocket), \( t \) is the time observed in the stationary frame (Earth), \( v \) is the relative velocity, and \( c \) is the speed of light.

Step 2: Setting the Time Dilated Condition

We want the time in the rocket, \( t' \), to be half of the time on Earth, \( t \). Therefore, we set \[ t' = \frac{t}{2} \]Substituting into the time dilation formula gives: \[ \frac{t}{2} = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \] Simplifying, we get \[ \sqrt{1 - \frac{v^2}{c^2}} = \frac{1}{2} \]

Step 3: Solving for Velocity

To find \( v \), square both sides of the equation:\[ 1 - \frac{v^2}{c^2} = \frac{1}{4} \]Rearrange the equation: \[ \frac{v^2}{c^2} = 1 - \frac{1}{4} = \frac{3}{4} \] Solve for \( v^2 \): \[ v^2 = \frac{3}{4}c^2 \]Taking the square root gives: \[ v = c \times \sqrt{\frac{3}{4}} = c \times \frac{\sqrt{3}}{2} \]

Step 4: Substituting the Speed of Light

Substitute the speed of light, \( c \approx 3 \times 10^8 \) meters per second, into the equation to find \( v \): \[ v = 3 \times 10^8 \times \frac{\sqrt{3}}{2} \approx 2.598 \times 10^8 \text{ m/s} \]

Step 5: Analyzing Jet Plane Speeds

Present-day jet planes usually fly at speeds around 250 to 900 meters per second, which is much less than the calculated value of \( 2.598 \times 10^8 \) meters per second. Hence, they are nowhere near achieving such speeds.

Key Concepts

Einstein's Theory of Relativity

Einstein's Theory of Relativity is a groundbreaking scientific theory that revolutionized our understanding of time, space, and motion. It has two main components: the Special Theory of Relativity and the General Theory of Relativity. In the context of time dilation, we focus on the Special Theory of Relativity.

One of the fundamental ideas in this theory is that the laws of physics are the same for all observers, regardless of their relative motion. This concept introduces the idea that time is not absolute, and can differ for observers moving at different velocities. According to the theory, as an object approaches the speed of light, time begins to slow down relative to a stationary observer. This phenomenon is known as time dilation.

• Time dilation occurs because the speed of light is constant in all frames of reference.
• The faster an object moves, the more noticeable the time slows down for it, compared to a stationary observer.

Understanding these principles allows us to see why time dilation becomes significant at high velocities close to the speed of light.

Relative Velocity

Relative velocity refers to the velocity of an object or observer A in the rest frame of another object or observer B. In the context of Einstein's relativity, it becomes crucial when considering observers in different frames of reference.

For instance, imagine a rocket moving away from Earth. The relative velocity of the rocket with respect to an observer on Earth will influence how time is perceived on the rocket compared to Earth. Time dilation effects become more prominent as this relative velocity approaches the speed of light. The equation used to calculate these effects is:

• \[ t' = \frac{t}{\sqrt{1 - \frac{v^2}{c^2}}} \]
• Here, \( v \) represents the relative velocity between the two observers.

This equation shows that as the relative velocity \( v \) increases, the factor \( \sqrt{1 - \frac{v^2}{c^2}} \) becomes smaller, magnifying the time dilation effect.
Understanding relative velocity is key to grasping time dilation and relativistic effects.

Speed of Light

The speed of light, denoted as \( c \), is a fundamental constant in physics and plays a pivotal role in relativity. Its value is approximately \( 3 \times 10^8 \) meters per second. In Einstein's theory, nothing can surpass the speed of light, making it a universal speed limit.

Relativity teaches us that the speed of light is the same for all observers, regardless of their state of motion. This constancy leads to many intriguing consequences, such as time dilation. It ensures that the closer an object's speed gets to the speed of light, the slower time appears to pass for that object compared to a stationary observer.

• Light travels at this incredible speed across vast distances in the universe.
• As objects move faster, towards light speed, their mass effectively increases, requiring more energy for further acceleration.

These concepts show the unique effects that the speed of light has on time and motion, underpinning the phenomena described by the Theory of Relativity.

Jet Plane Speeds

Jet plane speeds are significantly slower than the speed of light, which is the domain where relativistic effects like time dilation become observable. Typical jet planes cruise at speeds ranging from 250 to 900 meters per second.

Compared to the immense speed required to witness substantial time dilation—roughly 75% the speed of light, or \( 2.598 \times 10^8 \) meters per second—jet planes are nowhere near fast enough. Therefore, in practical terms, time dilation effects are negligible at these speeds and everyday aviation.

• For aviation, speeds are calculated in terms of Mach numbers, which are fractions of the speed of sound, not light.
• Although faster jets exist, they are still not capable of reaching speeds where relativistic time dilation would be significant.

Thus, while jet planes showcase some of the marvels of modern engineering, their speeds do not allow us to experience the astonishing effects of relativity observable at velocities approaching the speed of light.