Question

Bob is on Earth. Anna is on a spacecraft moving away from Earth at \(0.6 c .\) At some point in Anna's outward travel. Bob fires a projectile loaded with supplies out to Anna's ship. Relative to Bob, the projectile moves at \(0.8 c\). (a) How fast does the projectile move relative to Anna? (b) Bob also sends a light signal, "Greetings from Earth" out to Anna's ship. How fast does the light signal move relative to Anna?

Short answer

The speed of the projectile according to Anna is \(0.946c\). The speed of the light signal according to Anna is \(c\).

Step-by-step solution

Identify variables

Let's identify the given variables. Relative speed of Anna compared to Bob is \(0.6c\), where \(c\) is the speed of light. Relative speed of the projectile compared to Bob is \(0.8c\). We need to find the relative speed of the projectile compared to Anna, let's denote it as \(v\).

Apply the relativity addition formula to find the speed of the projectile relative to Anna

The formula is given by \(v = (v1 + v2) / (1 + (v1*v2 / c^2))\), where \(v1\) is the velocity of Anna and \(v2\) is the velocity of the projectile both with respect to Bob. Substituting the given values, we get \(v = (0.6c + 0.8c) / (1 + (0.6c*0.8c / c^2))\).

Simplify the expression

Simplifying this, we get \(v = 1.4c / (1 + 0.48) = 1.4c / 1.48 = 0.946c\). Hence, the speed of the projectile relative to Anna is \(0.946c\).

Find the speed of the light signal relative to Anna

In any reference frame, the speed of light is always \(c\), regardless of the motion of the source. So, the speed of the light signal relative to Anna would still be \(c\).

Key concepts

Velocity Addition Formula

In the realm of Special Relativity, traditional methods of adding velocities quickly become outdated. Unlike everyday experiences where speeds simply add up, Special Relativity introduces the Velocity Addition Formula. This formula is crucial when dealing with objects moving at speeds close to the speed of light, where the effects of relativity become significant. Instead of a simple addition, we use:
\[ v = \frac{v1 + v2}{1 + \left( \frac{v1 \cdot v2}{c^2} \right)} \]This equation accounts for the effects of relativity, ensuring that no object's speed ever exceeds the speed of light, \(c\). By using this formula, we can accurately determine the speed of an object as perceived from different reference frames.
For instance, in the exercise, Anna moves away from Earth at \(0.6c\), while Bob sends a projectile at \(0.8c\). To find the speed relative to Anna, simply plug these values into the Velocity Addition Formula. The output is not the sum of speeds, but rather a speed adjusted for relativistic effects, which in this case is \(0.946c\). This shows how relativistic physics transcends classical assumptions.

Speed of Light

The speed of light, denoted by \(c\), stands as a core pillar in the framework of Special Relativity. It is universally constant at approximately \(3 \times 10^8 \) meters per second and behaves uniquely compared to other velocities. In many physics problems, like the one involving Bob and Anna, the constancy of the speed of light leads to unexpected but profound insights.
When Bob sends a light signal to Anna, regardless of her motion or Bob's motion, Anna observes the speed of light to be \(c\). This defies our everyday intuitive understanding where velocities could simply add up. This invariance is a testament to Einstein's groundbreaking work, establishing that the speed of light is the same in all reference frames, even if those frames themselves are moving at significant fractions of the speed of light.
Understanding this principle helps to appreciate how the Universe maintains cosmic speed limits, ensuring no information or material object exceeds this fundamental speed.

Reference Frames

In the study of relativity, the concept of reference frames is crucial to understand how motion is perceived differently by different observers. A reference frame is essentially a perspective or viewpoint, often defined as an "inertial frame," from which motion is measured and observed.
For Anna, who is on a spacecraft traveling at \(0.6c\) away from Bob who is on Earth, her reference frame moves with her, affecting her perception of motion. Bob is in another reference frame, typically considered stationary for calculation simplicity, which affects how he perceives the projectile's velocity compared to his own.
In such relative scenarios, calculations must adopt these differing perspectives to make sense of speeds like Anna's observed projectile speed of \(0.946c\), differing from Bob's direct observation. Each reference frame follows its own path of perceiving time and distance, leading to the intriguing concepts explored in Einstein's theory.