I tried India's Hardest Exam

ZPhysics
26 Feb 202310:48
EducationalLearning
32 Likes 10 Comments

TLDRIn this challenging physics problem, the presenter tackles a question from the GRE Advanced exam involving two stars, A and B, with star B initially twice as massive as A. After an astrophysical event causes B to consume A, star A's radius shrinks to half, and its remaining mass forms a shell around B. The presenter calculates the escape velocities for both stars, deriving a formula to find the ratio of their velocities, ultimately determining the value of n to be 2.3. The detailed explanation showcases the complex process of solving a high-level physics problem, engaging viewers with its depth and clarity.

Takeaways
  • 📚 The video discusses a challenging physics problem from the GRE Advanced exam, which is known for its difficulty.
  • 🌟 The problem involves two stars, A and B, with star B initially having twice the mass of star A and different densities.
  • 🔄 An astrophysical event causes star B to consume star A, leading to star A shrinking and its remaining mass forming a shell around star B with the same density as star A.
  • 📉 Star A's new mass is calculated to be one-eighth of its original mass after the consumption process.
  • 📏 The new radius of star A is half of its original radius, and the density remains constant.
  • 🧭 The escape velocity formula is re-derived in the video, emphasizing its importance for solving the problem.
  • ⚖️ The escape velocity of star A (VA) is calculated using the formula and the new mass and radius of star A.
  • 🌌 For star B, the escape velocity (VB) is calculated considering the new mass after consuming star A and the original radius.
  • 🔢 The mass of star B after the event is the original mass plus seven-eighths of star A's original mass.
  • 📐 The new radius of star B is determined by the volume of the spherical shell formed by the remaining mass of star A.
  • 🔄 The ratio of the escape velocities (VB/VA) is simplified to find the value of 'n', which is the final answer to the problem.
  • 🎯 The final calculation reveals that 'n' equals 2.3, solving the GRE Advanced physics problem presented in the video.
Q & A

  • What is the context of the physics exam mentioned in the script?

    -The context is the General Advanced Physics exam, which is considered one of the most challenging physics exams in the world.

  • What is the initial condition of Star A and Star B in terms of mass and radius?

    -Initially, both Star A and Star B have the same radius, denoted as R. The mass of Star B is initially twice the mass of Star A.

  • What happens to Star A due to an astrophysical process?

    -Due to an astrophysical process, Star B starts devouring Star A, causing Star A to shrink to half of its original radius.

  • How does the density of Star A change after the process?

    -The density of Star A remains the same after the process, even though its volume has decreased.

  • What is the new mass of Star A after the process, and how is it related to its original mass?

    -The new mass of Star A, denoted as m_a dash, is one-eighth of its original mass, m_a.

  • What formula is used to calculate the escape velocity of a star?

    -The escape velocity formula used in the script is derived from setting the kinetic energy equal to the gravitational potential energy at the star's surface.

  • How is the escape velocity of Star A, VA, calculated in the script?

    -VA is calculated by setting the kinetic energy (1/2 m v^2) equal to the gravitational potential energy (GMm/R), where M is the mass, m is the reduced mass of Star A, and R is the new radius.

  • What is the new mass of Star B after devouring Star A?

    -The new mass of Star B is the original mass of Star B plus 7/8 of the original mass of Star A.

  • How does the volume of the shell around Star B relate to the original volume of Star A?

    -The volume of the shell is the difference between the volume of the new Star B and the original volume of Star A.

  • What is the final expression for the ratio of the escape velocities of Star B to Star A (VB/VA)?

    -The final expression for the ratio VB/VA is derived to be 23/(cube root of 15), indicating that n equals 2.3.

  • What conclusion is reached regarding the value of n in the given expression for the ratio of escape velocities?

    -The conclusion is that n equals 2.3, based on the derived expression for the ratio of the escape velocities of the two stars.

Outlines
00:00
🌌 Astrophysical Star Devouring Problem

This paragraph introduces a complex physics problem involving two stars, A and B, where star B has twice the mass of star A. Star A shrinks to half its radius after being consumed by star B, while maintaining its density. The problem requires calculating the escape velocity ratio (Vb/Va) of the stars using the given densities and masses. The solution involves deriving the new mass of star A and using the escape velocity formula, which is re-derived for clarity. The process includes setting up kinetic and gravitational energy equations and simplifying them to find the escape velocities for both stars.

05:02
🔍 Calculating Escape Velocities and Star Radii

The second paragraph delves into the calculations of the escape velocities for both stars after the astronomical event. It starts by determining the new mass of star A and its escape velocity using the rearranged kinetic and gravitational energy equations. The process then moves on to calculate the escape velocity for star B, considering the increase in its mass due to consuming star A. The key to solving this part of the problem is understanding the volume and density of the spherical shell formed around star B, which leads to finding the new radius of star B. The calculations are meticulous, involving algebraic manipulations to isolate and solve for the variables.

10:08
🎯 Solving for the Escape Velocity Ratio and Final Answer

The final paragraph wraps up the problem by calculating the ratio of the escape velocities (Vb/Va). It begins with simplifying the expressions for the escape velocities of both stars and then finding the ratio. The solution process involves careful algebraic manipulation to cancel out terms and isolate the variable n, which represents the power in the given expression for the escape velocity ratio. After significant work, the paragraph concludes with the discovery that n equals 2.3, solving the problem and providing a sense of satisfaction in tackling a challenging physics question.

Mindmap
Keywords
💡Physics exam
The term 'Physics exam' refers to a test or assessment that evaluates a student's knowledge and understanding of physics concepts. In the context of the video, it is specifically about a challenging exam known as the 'G Advanced,' which is implied to be a high-level physics examination. The script mentions the difficulty of the questions and the time constraints faced by candidates during the real exam.
💡Stars
In astronomy, 'stars' are celestial bodies composed mostly of hydrogen and helium, which emit light due to nuclear reactions in their cores. The video script discusses two stars, 'star A' and 'star B,' with different initial conditions and masses. The concept of stars is central to the problem presented, as the script explores their physical properties and interactions.
💡Density
Density is a measure of mass per unit volume, typically expressed in kilograms per cubic meter (kg/m^3). In the script, the densities of the two stars, denoted as 'rho A' and 'rho B,' are given and play a crucial role in the problem-solving process. The script explains how density is used to relate the mass and volume of the stars.
💡Astrophysical process
An 'astrophysical process' refers to any physical process that occurs in the universe, often involving the interaction of celestial bodies. In the video, 'star B' is described as 'devouring' 'star A,' which is an astrophysical process that leads to changes in the stars' masses and radii.
💡Escape velocity
Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body without further propulsion. The script requires finding the ratio of the escape velocities of the two stars, 'VB' over 'VA,' which is a key part of the problem.
💡Spherical shell
A 'spherical shell' in the context of the video refers to a layer around 'star B' that contains the remaining mass of 'star A' after it has been devoured. The density of this shell is given as 'rho A,' and the script uses this information to calculate the volume and, subsequently, the radius of the shell.
💡Volume
Volume is the quantity of three-dimensional space enclosed by a closed surface, often measured in cubic meters (m^3). The script discusses the volume of the stars and the spherical shell, using it to relate to their densities and masses.
💡Mass
Mass is a measure of the amount of matter in an object, typically measured in kilograms (kg). In the script, the initial and altered masses of the stars are central to the problem, with 'star B' initially having twice the mass of 'star A' and later acquiring additional mass from 'star A'.
💡Gravitational constant
The 'gravitational constant,' denoted as 'G,' is a physical constant involved in the calculation of gravitational force between two masses. In the script, 'G' is used in the formulae to calculate the escape velocities of the stars.
💡Ratio
A 'ratio' is a mathematical expression that compares two quantities, often expressed as a fraction or with a colon. The script aims to find the ratio of the escape velocities of the two stars, 'VB/VA,' which is a critical part of solving the problem.
💡Cube root
The 'cube root' of a number is a value that, when multiplied by itself three times, gives the original number. In the script, the cube root is used in the calculation of the new radius of 'star B' after the interaction with 'star A'.
Highlights

Introduction to the challenging G Advanced Physics exam, emphasizing the time constraint of two minutes per question.

Description of the problem involving two stars with initial equal radii and different densities, with Star B having twice the mass of Star A.

Explanation of an astrophysical process where Star B starts devouring Star A, causing Star A to shrink and its mass to redistribute.

Derivation of the new mass of Star A after the process, which is one-eighth of its original mass.

Calculation of the escape velocity for Star A using the formula for kinetic energy set equal to gravitational potential energy.

Introduction of the formula for escape velocity and the recommendation to memorize it for quick recall during exams.

Rederivation of the escape velocity formula for Star A, taking into account the new mass and radius.

Calculation of the escape velocity for Star B, considering the new mass after devouring Star A and the original radius.

Explanation of the mass change in Star B after the devouring process, including the acquisition of mass from Star A.

Derivation of the new radius of Star B using the density of the spherical shell formed after the process.

Calculation of the volume of the shell and its relation to the new radius of Star B.

Final calculation of the escape velocity for Star B using the new radius and adjusted mass.

Determination of the ratio of the escape velocities of Star B to Star A (VB/VA) and the expression for n.

Simplification of the ratio expression, leading to the conclusion that n equals 2.3.

Reflection on the problem-solving process and the satisfaction derived from solving complex physics problems.

Encouragement for viewers to explore more problems from the G Advanced exam for further practice.

Transcripts

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Physics ProblemEscape VelocityStar DevouringAstrophysicsDensity CalculationMass RatioVolume ShrinkageGalactic ProcessesEducational ContentProblem Solving