Problem 17 - University Physics with Modern Physics

Question

A pursuit spacecraft from the planet Tatooine is attempting to catch up with a Trade Federation cruiser. As measured by an observer on Tatooine, the cruiser is traveling away from the planet with a speed of 0.600 \(\mathrm{c}\) . The pursuit ship is traveling at a speed of 0.800 \(\mathrm{c}\) relative to Tatooine, in the same direction as the cruiser. (a) For the pursuit ship to catch the cruiser, should the speed of the cruiser relative to the pursuit ship be positive or negative? (b) What is the speed of the cruiser relative to the pursuit ship?

Step-by-Step Solution

Answer

(a) Negative (b) Approximately -0.385c.

Step 1: Understanding Relative Motion

We want to find the speed of the cruiser as observed from the pursuit spacecraft. For part (a), we should consider that if the pursuit ship is faster, then relative to the pursuit ship, the cruiser is moving in the opposite direction. Therefore, in such a case, the relative speed would be negative. Conversely, if the pursuit ship were slower, which is not the case here, the relative speed would be positive.

Step 2: Using the Relativistic Velocity Addition Formula

To solve part (b), we use the relativistic velocity addition formula:\[ v' = \frac{v - u}{1 - \frac{vu}{c^2}} \]where \( v = 0.800c \) is the speed of the pursuit ship, \( u = 0.600c \) is the speed of the cruiser, and \( c \) is the speed of light.

Step 3: Substituting the Values

Substitute the known values into the formula:\[ v' = \frac{0.800c - 0.600c}{1 - \frac{(0.800c)(0.600c)}{c^2}} \]

Step 4: Calculating the Numerator

Calculate the numerator:\[ 0.800c - 0.600c = 0.200c \]

Step 5: Calculating the Denominator

Calculate the denominator:\[ 1 - \frac{(0.800)(0.600)c^2}{c^2} = 1 - 0.480 = 0.520 \]

Step 6: Final Calculation

Now perform the final calculation:\[ v' = \frac{0.200c}{0.520} \approx 0.385c \]So, the speed of the cruiser relative to the pursuit ship is approximately \(-0.385c\), indicating that from the perspective of the pursuit ship, the cruiser appears to move backwards at this speed.

Key Concepts

Relative Motion

When we talk about relative motion, we are focusing on how the speed of one object appears to another observer. This isn't as straightforward as it seems, especially when dealing with high velocities close to the speed of light. In the context of high speeds, such as a spacecraft racing after another, we need to consider how different observers view their relative velocities. Here, since the pursuit ship is faster, it perceives the cruiser as moving in the opposite direction. Therefore, for the pursuit spacecraft, the relative velocity of the cruiser is negative. This is simply because the pursuit is closing the gap as it moves faster than the cruiser.

Special Relativity

Special relativity is a theory coined by Albert Einstein, which radically transformed our understanding of motion, especially when objects move near the speed of light. One of its most mind-boggling concepts is that time and space are not absolute, and they vary depending on the observer's speed. When objects travel at a significant fraction of the speed of light, relativistic effects become essential. These effects include how distances, time intervals, and even the order of events can appear different to observers in different inertial frames.

Velocity Transformation

Velocity transformation is a method within special relativity to calculate how quickly an object appears to move relative to another moving reference frame. The key tool for this is the relativistic velocity addition formula:

• It ensures that no object exceeds the speed of light irrespective of the observers' frame of reference.
• Under this formula, velocities do not add up the same way they do at lower velocities.

The formula used here, \[ v' = \frac{v - u}{1 - \frac{vu}{c^2}} \]is specifically designed for combining velocities that are along the same line. It accounts for the relativistic effects by adjusting the sum of velocities as perceived by a stationary observer.

Speed of Light

The speed of light, often denoted as \( c \), is a fundamental constant of nature, approximately valued at \( 3 \times 10^8 \) meters per second. It's the ultimate speed limit in the universe, dictating not only the flow of information, but also shaping the very fabric of time and space as we perceive it. No material object can reach, let alone surpass, this speed.

• In relativity, \( c \) serves as a natural speed boundary.
• Its constancy ensures equivalence in the laws of physics across all frames of reference.

The concept of light's speed touches on deep principles about the universe's structure, influencing how we calculate velocities and ensure nothing violates this cosmic speed limit.