Problem 64 - University Physics with Modern Physics

Question

Rocket A passes Earth at a speed of \(0.75 c\). At the same time, rocket B passes Earth moving \(0.95 c\) relative to Earth in the same direction. How fast is B moving relative to A when it passes \(A ?\)

Step-by-Step Solution

Answer

Answer: The relative speed of Rocket B with respect to Rocket A is approximately 0.9926c.

Step 1: 1. Understand the relativistic velocity addition formula

The relativistic velocity addition formula is given by: $$ V = \frac{u+v}{1+\frac{uv}{c^2}} $$ where \(V\) is the relative speed of the two objects, \(u\) is one object's speed relative to a reference frame (in this case, Earth), \(v\) is the other object's speed relative to the same reference frame, and \(c\) is the speed of light.

Step 2: 2. Substitute the given values into the formula

In our problem, the speed of Rocket A relative to Earth is \(u = 0.75c\), and the speed of Rocket B relative to Earth is \(v = 0.95c\). We'll now substitute these values into the relativistic velocity addition formula: $$ V = \frac{(0.75c) + (0.95c)}{1 + \frac{(0.75c)(0.95c)}{c^2}} $$

Step 3: 3. Simplify the expression

We can simplify the expression by factoring out \(c\) in the numerator and canceling out with the \(c^2\) in the denominator: $$ V = \frac{c(0.75 + 0.95)}{1 + \frac{0.75 \cdot 0.95}{1}} $$

Step 4: 4. Calculate the relative speed

Now, we can compute the relative speed by solving for V: $$ V = \frac{c(1.7)}{1 + 0.7125} $$ $$ V = \frac{1.7c}{1.7125} $$ $$ V \approx 0.9926c $$ So, Rocket B is moving at a speed of approximately \(0.9926c\) relative to Rocket A when it passes Rocket A.

Key Concepts

Special Relativity

Special relativity is a fundamental theory proposed by Albert Einstein in 1905 to address inconsistencies between classical mechanics and electromagnetic theory. The theory fundamentally altered our understanding of time and space, introducing the concept that the laws of physics are the same for all non-accelerating observers, and that the speed of light in vacuum is constant regardless of the observer's motion or the motion of the source.

• Special relativity introduces the idea that time and space are interconnected in a four-dimensional spacetime continuum.
• This concept leads to phenomena such as time dilation and length contraction, where time could seem to pass slower, and lengths can appear contracted depending on the relative speed of the observer.

These ideas are crucial when dealing with high-speed scenarios, like those approaching the speed of light, which cannot be accurately described by Newtonian physics. When velocities are high, such as in this problem with rockets moving at significant fractions of the speed of light, special relativity must be used to accurately calculate velocities.

Speed of Light

The speed of light, denoted as \(c\), is one of the key constants in physics, valued at approximately \(299,792,458\) meters per second. It is not only a measure of how fast light moves through a vacuum, but it also sets a fundamental speed limit for the universe. No object with mass can achieve or exceed this speed, because as an object's speed approaches the speed of light, its mass effectively becomes infinite, and it would require infinite energy to move faster.

• The speed of light is vital in calculations in special relativity, particularly when applying the relativistic velocity addition formula.
• In the problem of rockets, the speeds are given as fractions of \(c\), such as \(0.75c\) and \(0.95c\).

These fractions indicate how close objects are traveling relative to the speed of light and highlight the importance of using relativistic physics to solve the problem as opposed to classical mechanics.

Relative Motion

Relative motion refers to the calculation of one object's motion from the viewpoint of another object, often a crucial concept in scenarios involving high-speed travel, like spacecrafts and rockets. In the context of special relativity, relative motion takes on a complex nature because velocities that are significant fractions of the speed of light need to be combined using the relativistic velocity addition formula, instead of simply being added together.

• The formula \(V = \frac{u+v}{1+\frac{uv}{c^2}}\) accounts for the effects of special relativity when determining the relative velocity.
• The expression resolves the issue of combining speeds in a way that aligns with the constancy of the speed of light.

In the exercise, Rocket B's speed relative to Rocket A was calculated by considering each rocket's velocity relative to Earth and using this formula. As the speeds approach the speed of light, significant relativistic effects are considered, and simple arithmetic addition just won't suffice, ensuring the results obey the universal speed limit set by the speed of light.