I E Irodov Physical Fundamentals Of Mechanics (Relativistic Mechanics) Q. 1.388 Solution

I E Irodov Physical Fundamentals of Mechanics has become a popular advanced Physics book for students who are preparing for JEE Advanced, Physics Olympiads and other competitive exams.

The Relativistic Mechanics section deals with more advanced topics like relativistic momentum, mass-energy relations, velocity transformation and motion at speeds close to the speed of light.

On this page you will find detailed step by step solution for I E Irodov Physical Fundamentals Of Mechanics, Relativistic Mechanics Question 1.388 with concepts and formulas used in the solution.

I E Irodov Relativistic Mechanics Question 1.388

Question:

A relativistic rocket ejects a gas jet with a velocity (u), constant relative to the rocket, but non relativistic.

Find how the velocity (v) of the rocket depends on its rest mass (m), if the initial rest mass of the rocket is (m_0).

Concepts Used in I E Irodov Q. 1.388

This problem is based on relativistic rocket motion.

Important concepts used include:

• Conservation of relativistic momentum
• Conservation of energy
• Relativistic velocity
• Rocket mass loss
• Rapidity
• Relativistic rocket equation

We assume the exhaust velocity (u) is constant relative to the rocket. It is also a lot less than the speed of light.

Easy Solution of I E Irodov Relativistic Mechanics Q. 1.388

Let:

• (m_0) = initial rest mass of the rocket
• (m) = rest mass of the rocket at any later time
• (u) = velocity of the emitted gas relative to the rocket
• (v) = velocity of the rocket
• (c) = speed of light

We need to find the relation between (v) and (m).

Step 1: Use the Relativistic Rocket Relation

For a rocket moving at relativistic speed, it is convenient to express velocity using rapidity.

The rapidity (\eta) is defined by:

[
\frac{v}{c}=\tanh \eta
]

For a small loss in rocket mass, the change in rapidity is:

[
d\eta=-\frac{u}{c}\frac{dm}{m}
]

The negative sign appears because the rocket mass decreases while its velocity increases.

Step 2: Integrate the Equation

Initially:

[
m=m_0
]

and the rocket is assumed to start from rest:

[
v=0
]

Therefore:

[
\eta=0
]

At a later stage, when the rocket mass becomes (m), integrate:

-\frac{u}{c}\int_{m_0}^{m}\frac{dm}{m}
]

This gives:

-\frac{u}{c}\ln\left(\frac{m}{m_0}\right)
]

Using the logarithm property:

\frac{u}{c}\ln\left(\frac{m_0}{m}\right)
]

Step 3: Convert Rapidity into Rocket Velocity

We know:

[
\frac{v}{c}=\tanh \eta
]

Substituting the value of (\eta):

\tanh\left[
\frac{u}{c}\ln\left(\frac{m_0}{m}\right)
\right]
]

Therefore:

c\tanh\left[
\frac{u}{c}\ln\left(\frac{m_0}{m}\right)
\right]
]

Final Answer

The velocity of the relativistic rocket as a function of its remaining rest mass is:

c\tanh\left[
\frac{u}{c}\ln\left(\frac{m_0}{m}\right)
\right]
}
]

Here:

• (m_0) is the initial rest mass of the rocket
• (m) is its remaining rest mass
• (u) is the gas exhaust velocity relative to the rocket
• (c) is the speed of light

Alternative Form of the Answer

Using the identity:

[
\tanh x=\frac{e^{2x}-1}{e^{2x}+1}
]

the velocity can also be written as:

c\frac{
\left(\frac{m_0}{m}\right)^{2u/c}-1
}{
\left(\frac{m_0}{m}\right)^{2u/c}+1
}
}
]

Both expressions represent the same relativistic rocket equation.

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Simple Explanation of the Answer

The rocket releases gas , so its rest mass decreases from ( m_0 ) to ( m ) .

The rocket moves forward because the exhaust gases move backward. In traditional mechanics, the velocity of a rocket grows logarithmically with the mass ratio.

But at relativistic speeds the maximum velocity of the rocket can not be higher than the speed of light. So the hyperbolic tangent function shows up in the final expression:

[
v=c\tanh(\text{quantity})
]

Since the value of (\tanh x) always remains less than 1:

[
v<c
]

This ensures that the rocket never exceeds the speed of light.

Key Takeaways

Common Mistakes Students Should Avoid

Students often make these mistakes while solving I E Irodov Q. 1.388:

• Using the classical rocket equation directly at relativistic speeds
• Forgetting that the rocket velocity must remain below (c)
• Confusing the exhaust velocity (u) with the rocket velocity (v)
• Using (\ln(m/m_0)) instead of (\ln(m_0/m))
• Missing the factor (u/c)
• Forgetting to convert rapidity back into velocity using the hyperbolic tangent

Students should carefully identify the initial mass, remaining mass and reference frame of the gas velocity.

Why Practice I E Irodov Problems for JEE Advanced?

I E Irodov problems help students develop a deeper understanding of advanced Physics.

Benefits include:

• Strong conceptual understanding
• Better mathematical application
• Advanced problem-solving ability
• Improved analytical thinking
• Preparation for challenging Physics questions
• Greater confidence for JEE Advanced

Students should solve selected I E Irodov problems only after completing the relevant concepts and standard JEE-level questions.

Is I E Irodov Q. 1.388 a JEE Main Question?

No. I E Irodov Q. 1.388 is not a typical JEE Main-level question.

The problem uses advanced relativistic mechanics and the concept of rapidity. It is more suitable for:

• Advanced Physics practice
• JEE Advanced aspirants
• Physics Olympiad preparation
• Students interested in relativistic mechanics

For JEE Main, students should prioritize NCERT concepts, the official syllabus, previous year questions and mock tests.

Is Relativistic Mechanics Important for JEE?

Relativistic mechanics is not usually treated as a major high-weightage chapter for JEE Main.

However, selected relativity concepts can help advanced students strengthen their understanding of:

• Special relativity
• Relativistic momentum
• Mass-energy relationship
• Velocity transformation
• Length contraction
• Time dilation

Students should study this topic after completing the core JEE Physics syllabus.

Important Topics in I E Irodov Physical Fundamentals of Mechanics

How Should Students Use I E Irodov for JEE Advanced?

Recommended preparation order:

Students should not try to solve every I E Irodov question without understanding the underlying theory.

I E Irodov for NRI JEE Students

Selected I E Irodov questions for advanced Physics practice for JEE preparation for NRI students from USA, UAE, Saudi Arabia, Qatar, Oman, Kuwait and Bahrain.

Students studying under AP, IB, American or international school curricula should first understand the difference between their school syllabus and JEE syllabus.

A suitable preparation approach is:

• Complete the JEE Physics syllabus
• Solve chapter-wise JEE questions
• Practice previous year papers
• Attempt JEE Advanced problems
• Use I E Irodov for selected advanced topics

Online JEE coaching can help NRI students select the correct problems instead of spending time on questions outside their preparation requirements.

Related I E Irodov Relativistic Mechanics Questions

Students can also practice:

• I E Irodov Relativistic Mechanics Q. 1.345
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• I E Irodov Relativistic Mechanics Q. 1.359
• I E Irodov Relativistic Mechanics Q. 1.378
• I E Irodov Relativistic Mechanics Q. 1.388

Link each question to its corresponding TestPrepKart solution page.

Practice Next

After completing I E Irodov Q. 1.388, students can continue with:

• JEE Advanced Physics revision notes
• JEE previous year question papers
• Relativistic mechanics practice questions
• Conservation of momentum questions
• Advanced mechanics problems
• JEE Advanced mock tests

Useful JEE, DASA & CIWG Resources for I E Irodov Physics

FAQs on I E Irodov Relativistic Mechanics Q. 1.388

1. What is the answer to I E Irodov Q. 1.388?

The rocket velocity is:

c\tanh\left[
\frac{u}{c}\ln\left(\frac{m_0}{m}\right)
\right]
]

2. Which concept is used in I E Irodov Q. 1.388?

The problem uses relativistic rocket motion, conservation principles and rapidity.

3. Is I E Irodov Q. 1.388 useful for JEE Main?

It is above the usual JEE Main level. It is more suitable for advanced Physics and JEE Advanced practice.

4. Why is the hyperbolic tangent used in the answer?

The hyperbolic tangent ensures that the velocity of the rocket always remains below the speed of light.

5. What does (m_0/m) represent?

It is the rocket’s mass ratio:

[
\frac{\text{initial rest mass}}{\text{remaining rest mass}}
]

A larger mass ratio produces a higher rocket velocity.

6. Can the rocket reach the speed of light?

No. The rocket velocity approaches the speed of light but cannot reach or exceed it.

7. Is I E Irodov useful for JEE Advanced?

Selected I E Irodov questions are useful for students who have already completed standard JEE Advanced Physics preparation.