Question

A rocket traveling at speed $\frac{1}{2}c$ relative to frame $\mathcal{S}$ shoots forward bullets traveling at speed $\frac{3}{4}c$ relative to the rocket. What is the speed of the bullets relative to $\mathcal{S}?$

Short answer

The speed of the bullets relative to $\mathcal{S}$ is $\frac{10}{11}c$.

Step-by-step solution

Identify Given Information

We have a rocket traveling at speed $\frac{1}{2}c$ with respect to frame $\mathcal{S}$ and bullets traveling at speed $\frac{3}{4}c$ relative to the rocket. We need to find the speed of the bullets with respect to $\mathcal{S}$.

Apply the Relativistic Velocity Addition Formula

The speed of the bullets relative to $\mathcal{S}$ can be found using the relativistic velocity addition formula: $$u^{\prime }=\frac{u+v}{1+\frac{uv}{c^2}}$$where $u$ is the speed of the rocket, $v$ is the speed of the bullets relative to the rocket, and $c$ is the speed of light.

Substitute the Values

Substitute $u=\frac{1}{2}c$ and $v=\frac{3}{4}c$ into the formula:$$u^{\prime }=\frac{\frac{1}{2}c+\frac{3}{4}c}{1+\frac{\left(\frac{1}{2}c\right)\left(\frac{3}{4}c\right)}{c^2}}$$

Simplify the Numerator

Calculate the numerator:$$u^{\prime }=\frac{\frac{1}{2}c+\frac{3}{4}c}{1+\frac{3}{8}}=\frac{(\frac{1}{2}+\frac{3}{4})c}{1}$$Simplify $\frac{1}{2}+\frac{3}{4}$ to get $\frac{5}{4}$:$$u^{\prime }=\frac{\frac{5}{4}c}{1+\frac{3}{8}}$$

Simplify the Denominator

Calculate the denominator:$$1+\frac{3}{8}=\frac{8}{8}+\frac{3}{8}=\frac{11}{8}$$

Calculate the Final Speed

Substitute the simplified numerator and denominator back into the formula:$$u^{\prime }=\frac{\frac{5}{4}c}{\frac{11}{8}}$$Multiply by the reciprocal of the denominator:$$u^{\prime }=\frac{5}{4}\times \frac{8}{11}c=\frac{40}{44}c=\frac{10}{11}c$$

Final Answer

The speed of the bullets relative to frame $\mathcal{S}$ is $\frac{10}{11}c$.

Key concepts

Speed of Light

The speed of light, commonly denoted as $c$, is a fundamental constant of nature. It is the speed at which light travels in a vacuum, approximately $299,792,458$ meters per second or roughly $186,282$ miles per second. This value is crucial because it represents the ultimate speed limit in the universe.

In theories of physics, particularly relativity, the speed of light is a unique and unchanging figure. Whether you're observing a beam of light here on Earth or from a distant galaxy, light travels at this constant speed. This constancy forms the foundation for Albert Einstein's theory of relativity, influencing how we understand time and space.

When examining problems in physics, especially those involving high velocities like rockets or particles, the speed of light plays a vital role.

• It serves as the maximum speed that anything can travel.
• It affects time dilation, where time appears to slow down as an object approaches the speed of light.
• It enters into calculations through equations such as the relativistic velocity addition formula, which corrects for speeds approaching $c$.

Understanding the importance of the speed of light helps us comprehend the limitations and behavior of objects moving at significant fractions of this speed.

Relativity

Relativity is a principle developed by Albert Einstein, explaining how observers perceive space and time differently, depending on their relative motion. This theory consists of two main parts: Special Relativity and General Relativity.

Special Relativity applies to objects moving at constant speeds, particularly those approaching the speed of light. It introduces the concept that the laws of physics are the same for all observers in uniform motion. Here are some crucial ideas from special relativity:

• Time Dilation: Time intervals appear longer for objects moving at high speeds compared to stationary observers.
• Length Contraction: Objects appear shorter in the direction of motion from the viewpoint of a stationary observer.
• Mass Increase: Objects gain mass as they approach the speed of light.

General Relativity, on the other hand, deals with gravity's effect on space and time, predicting phenomena such as black holes and the bending of light around massive objects. However, special relativity is often at play when we discuss rocket physics and relativistic speed calculations.

In the context of the exercise, relativity allows us to understand why we cannot simply add velocities in the same straightforward way as with everyday speeds. Instead, the relativistic velocity addition formula is used, ensuring that the result never exceeds the speed of light, no matter how fast the components move.

Rocket Physics

Rocket physics involves understanding how rockets move and interact within the universe, often at speeds where relativistic effects cannot be ignored. In simple terms, rocket physics explores motion, force, and the properties of the rocket's travel through space.

One cornerstone of rocket physics is Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. In a rocket's context, gas expulsion downwards propels the rocket upwards. However, as a rocket reaches velocities close to the speed of light, classical physics transitions into relativistic physics.

For example, when studying a rocket firing bullets as in the exercise, the bullets' relative velocity needs careful consideration through Einstein's relativistic equations. Here, the relativistic velocity addition formula helps compute how speeds combine to stay consistent with the limitations imposed by the speed of light, avoiding any scenario where speeds might add up to exceed $c$.

Working through rocket physics involves:

• Calculating trajectories considering the impacts of relativity.
• Applying special relativity to understand energy and momentum at high velocities.
• Integrating time dilation and length contraction effects into practical applications.

Understanding rocket physics isn't just about getting rockets off the ground; it's about preparing for and navigating the complex, high-speed world of space travel.