Problem 19 - College 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\(c .\) The pursuit ship is traveling at a speed of 0.800\(c\) relative to Tatooine, in the same direction as the cruiser. What is the speed of the cruiser relative to the pur- suit ship?

Step-by-Step Solution

Answer

Relative speed is approximately 0.385c.

Step 1: Identify Given Values

We have the speed of the cruiser as 0.600\(c\) and the speed of the pursuit spacecraft as 0.800\(c\), both measured relative to the observer on Tatooine. The goal is to find the speed of the cruiser relative to the pursuit spacecraft.

Step 2: Know The Concept

We will use the relativistic velocity addition formula because the speeds are given as fractions of the speed of light \(c\). For two objects moving in the same direction, the formula is \( u' = \frac{u - v}{1 - \frac{uv}{c^2}} \), where \(u\) is the speed of the pursuit ship and \(v\) is the speed of the cruiser.

Step 3: Substitute Given Values into Formula

Substitute the given speeds from Step 1 into the formula: \( u' = \frac{0.800c - 0.600c}{1 - \frac{(0.800c)(0.600c)}{c^2}} \).

Step 4: Simplify the Expression

Simplify the numerator: \(0.800c - 0.600c = 0.200c\). Then simplify the denominator: \(1 - \frac{(0.800c)(0.600c)}{c^2} = 1 - 0.48 = 0.52\).

Step 5: Perform the Final Calculation

Now calculate the speed: \( u' = \frac{0.200c}{0.52} \). Simplify to find \( u' \approx 0.385c \).

Step 6: Conclusion

Thus, the speed of the cruiser relative to the pursuit spacecraft is approximately \(0.385c\).

Key Concepts

Speed of Light

The speed of light is a fundamental constant in physics and is denoted by the symbol \(c\). It is the speed at which light travels in a vacuum and has a value of approximately \(299,792,458\) meters per second. This is the maximum speed at which information or matter can travel in the universe, according to the principles of Einstein’s theory of special relativity.

One of the remarkable implications of the speed of light is that it appears the same to all observers, no matter their state of motion. This constancy is a cornerstone of special relativity, making processes that involve velocities close to \(c\) particularly interesting and complex.

It is important to understand that when we talk about speeds as fractions of \(c\), like \(0.600c\) or \(0.800c\), we are expressing these speeds as ratios of the speed of light. This notation simplifies calculation and helps when using formulas derived from relativistic physics, which are necessary when dealing with speeds at such high proportions of \(c\).

Relative Velocities

Relative velocities refer to the concept of measuring the speed of one object as observed from another object. In scenarios involving high speeds close to that of light, as in our problem with the cruiser and the pursuit spacecraft, standard arithmetic addition of velocities does not apply.

In such cases, we use the relativistic velocity addition formula: \[ u' = \frac{u - v}{1 - \frac{uv}{c^2}} \] - \(u\) is the speed of the pursuit ship relative to a stationary observer.- \(v\) is the speed of the cruiser relative to the same observer.- \(u'\) is the speed of the cruiser relative to the pursuit ship.

This formula adjusts for the effects of special relativity, ensuring that no resultant velocity exceeds the speed of light. Importantly, relativistic velocity addition accounts for how time and space are experienced differently by observers in different inertial frames of reference when they are moving close to the speed of light.

Special Relativity

Special relativity, introduced by Albert Einstein in 1905, fundamentally transformed our understanding of space, time, and velocity. It describes how measurements of space and time change for observers in different inertial frames, especially when they move at speeds close to the speed of light.

At its core, special relativity rests on two postulates:

• The laws of physics are the same in all inertial frames, meaning they do not change regardless of constant relative motion between observers.
• The speed of light in a vacuum is constant, measured at \(c\), regardless of the movement of the light source or the observer.

These postulates introduce counterintuitive concepts such as time dilation—where time stretches out at high velocities—and length contraction—where objects contract in the direction of motion.

In our problem, special relativity comes into play when calculating the speed of one ship relative to another. Classical physics' simple addition would not suffice; instead, we rely on Einstein’s formula for relativistic velocity addition. This ensures that speeds calculated remain true to the universal speed limit imposed by the constant \(c\).