Question

A rocket propels itself rectilinearly by giving portions of its mass a constant fbackward) velocity \(U\) relative to its instantaneous rest frame. It continues to do so until it attains a velocity \(V\) relative to its initial rest frame. Prove that the ratio of the initial to the final rest mass of the rocket is given by $$ \frac{M_{\mathrm{i}}}{M_{\mathrm{f}}}=\left(\frac{c+V}{c-V}\right)^{\frac{c}{2 U}} $$

Short answer

Answer: The ratio of the initial to the final rest mass of the rocket is given by $\left(\frac{c+V}{c-V}\right)^{\frac{c}{2U}}$.

Step-by-step solution

Understand the given data

We are given the backward velocity of the ejected mass \(U\) relative to the rocket's rest frame, the final velocity of the rocket \(V\), and the speed of light \(c\). Our job is to derive the ratio of the rocket's initial rest mass \(M_i\) to its final rest mass \(M_f\).

Define the velocity of ejected mass in the initial rest frame

To solve this problem, we first need to find the velocity of the ejected mass in the initial rest frame. According to the velocity addition formula in special relativity, the velocity of the ejected mass in the initial rest frame is given by: $$ w = \frac{U + V}{1 + \frac{UV}{c^2}} $$

Apply momentum conservation

For momentum conservation, the momentum of the ejected mass plus the final momentum of the rocket should be equal to the initial momentum of the rocket. Since the rocket is initially at rest, its momentum is 0. Thus, we can write the conservation of momentum as: $$ m_dw = (M_i - m_d) V $$ where \(m_d\) is the small mass ejected and \(w\) is the velocity of the ejected mass in the initial rest frame, as defined in Step 2.

Analyze the mass and energy simultaneous variation

As the rocket is giving away portions of its mass, the mass ejected \(m_d\) and the remaining mass can be related by the following differential equation: $$ \frac{dm_d}{M_i - m_d} = \frac{m_dw}{V (M_i - m_d)} $$

Integrate the differential equation

To find the mass ratio, we need to integrate the differential equation in Step 4. After integration, we obtain: $$ \ln \frac{M_i}{M_i - m_d} = -\frac{Vw}{c^2}\frac{M_i}{U} $$

Replace \(w\) with the given data

As we derived \(w\) in Step 2, we can replace the expression in the above equation and simplify it: $$ \ln \frac{M_i}{M_f} = \frac{(c+V)c}{2U} $$ where \(M_f = M_i - m_d\) is the final rest mass of the rocket.

Find the mass ratio

Now, we can find the mass ratio by exponentiating both sides of the equation in Step 6: $$ \frac{M_i}{M_f} = \left(\frac{c+V}{c-V}\right)^{\frac{c}{2U}} $$ So, the ratio of the initial to the final rest mass of the rocket is given by \(\left(\frac{c+V}{c-V}\right)^{\frac{c}{2U}}\).

Key concepts

Special Relativity

Special relativity is a fundamental theory in physics developed by Albert Einstein. It addresses the physics of objects moving at significant fractions of the speed of light. Unlike classical physics, it makes groundbreaking predictions about time, distance, and mass.
The theory introduces the concept that the laws of physics are the same in all inertial frames, meaning no preferential frames of reference. This leads to the famous equation:\( E=mc^2 \), which relates mass and energy. It also introduces the idea that time and space are intertwined in what is known as spacetime.
Special relativity is crucial for explaining high-speed phenomena that cannot be accurately described by Newtonian physics. It challenges our intuitive notions of absolute time and space by showing that they are relative and can change depending on an observer's state of motion.

Velocity Addition

When dealing with special relativity, adding velocities isn't straightforward like in classical physics. The concept of velocity addition must follow Einstein's postulates, ensuring that the speed of light remains constant across all frames of reference.
The formula for velocity addition in special relativity differs from the classical form. If an object moves with velocity \( U \) relative to another object, which itself moves at velocity \( V \) relative to the initial frame, the velocity \( w \) of the object relative to the initial frame is calculated by:

1. \[ w = \frac{U + V}{1 + \frac{UV}{c^2}} \]

This equation ensures that the resulting velocity \( w \) never exceeds the speed of light \( c \). Using this formula helps relate the velocities of different objects in various reference frames.

Momentum Conservation

Momentum conservation is a principle stating that the total momentum of an isolated system remains constant if no external forces act upon it. This principle holds true in both classical and relativistic physics, though its application in special relativity requires specific modifications due to mass-energy equivalence.
In the context of rockets, as mentioned in the step-by-step solution, the momentum of ejected mass and the rocket must balance. For a rocket at rest initially, its total momentum equals zero. As it expels part of its mass, the forward and backward forces create equal and opposite momenta, ensuring that net momentum stays zero. Consequently, the momentum of the ejected mass \( m_d \) moving with velocity \( w \) matches the final momentum of the rocket, which translates to:

1. \[ m_dw = (M_i - m_d) V \]

This equation elaborates how the distribution of mass and velocity affects the system's momentum.

Mass-Energy Equivalence

Mass-energy equivalence is a foundational concept in physics, expressed by the equation \( E=mc^2 \). It reveals that mass and energy are two sides of the same coin and can convert into one another.
In the context of rockets, as the rocket expels mass, it converts its stored chemical energy into kinetic energy, demonstrating mass-energy equivalence in action. When a piece of the rocket's mass becomes energy through propulsion, it shows matter turning into motion energy.
This principle also plays a significant role during the derivation of the initial to final rest mass ratio of the rocket. As mass decreases during propulsion, the corresponding energy manifests as the rocket’s increasing velocity. Applying this concept allows us to understand the profound connection between mass variation and energy conservation during rocket propulsion.