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Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth is
- a)8 : 1
- b)6 : 1
- c)7 : 1
- d)3 : 1
Correct answer is option 'D'. Can you explain this answer?
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Given:
Radius of the planet = 3 times the radius of the Earth
To find:
The ratio of escape speeds from the surface of the planet and the surface of the Earth
Formula:
The escape speed from the surface of a planet (v) can be given by the formula:
v = √(2GM/R)
where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.
Analysis:
1. Radius of the planet:
Let's assume the radius of the Earth is R and the radius of the planet is 3R.
2. Mass of the planet:
The average density of the planet is the same as that of the Earth. Since density = mass/volume, the mass of the planet can be expressed as:
Mass of the planet = Average density of the planet × Volume of the planet
The volume of a sphere is given by the formula:
Volume of a sphere = (4/3)π(r^3)
where r is the radius of the sphere.
Since the radius of the planet is 3R, the volume of the planet can be expressed as:
Volume of the planet = (4/3)π((3R)^3) = (4/3)π(27R^3) = 36πR^3
Therefore, the mass of the planet can be expressed as:
Mass of the planet = Average density of the Earth × 36πR^3
3. Escape speed from the surface of the planet:
Using the formula for escape speed, we can express the escape speed from the surface of the planet as:
vplanet = √(2G × Mass of the planet / Radius of the planet)
4. Escape speed from the surface of the Earth:
Using the formula for escape speed, we can express the escape speed from the surface of the Earth as:
vearth = √(2G × Mass of the Earth / Radius of the Earth)
Calculation:
Substituting the values of mass and radius of the planet and the Earth into the formulas for escape speed, we get:
vplanet = √(2G × Average density of the Earth × 36πR^3 / 3R)
vearth = √(2G × Average density of the Earth × 4πR^3 / R)
Simplifying the equations, we get:
vplanet = √(24GπR^2)
vearth = √(8GπR^2)
Taking the ratio of vplanet to vearth, we get:
vplanet/vearth = √(24GπR^2) / √(8GπR^2)
vplanet/vearth = √(24/8)
vplanet/vearth = √3
Therefore, the ratio of the escape speeds from the surface of the planet to the surface of the Earth is 3:1, which corresponds to option D.
Radius of the planet = 3 times the radius of the Earth
To find:
The ratio of escape speeds from the surface of the planet and the surface of the Earth
Formula:
The escape speed from the surface of a planet (v) can be given by the formula:
v = √(2GM/R)
where G is the gravitational constant, M is the mass of the planet, and R is the radius of the planet.
Analysis:
1. Radius of the planet:
Let's assume the radius of the Earth is R and the radius of the planet is 3R.
2. Mass of the planet:
The average density of the planet is the same as that of the Earth. Since density = mass/volume, the mass of the planet can be expressed as:
Mass of the planet = Average density of the planet × Volume of the planet
The volume of a sphere is given by the formula:
Volume of a sphere = (4/3)π(r^3)
where r is the radius of the sphere.
Since the radius of the planet is 3R, the volume of the planet can be expressed as:
Volume of the planet = (4/3)π((3R)^3) = (4/3)π(27R^3) = 36πR^3
Therefore, the mass of the planet can be expressed as:
Mass of the planet = Average density of the Earth × 36πR^3
3. Escape speed from the surface of the planet:
Using the formula for escape speed, we can express the escape speed from the surface of the planet as:
vplanet = √(2G × Mass of the planet / Radius of the planet)
4. Escape speed from the surface of the Earth:
Using the formula for escape speed, we can express the escape speed from the surface of the Earth as:
vearth = √(2G × Mass of the Earth / Radius of the Earth)
Calculation:
Substituting the values of mass and radius of the planet and the Earth into the formulas for escape speed, we get:
vplanet = √(2G × Average density of the Earth × 36πR^3 / 3R)
vearth = √(2G × Average density of the Earth × 4πR^3 / R)
Simplifying the equations, we get:
vplanet = √(24GπR^2)
vearth = √(8GπR^2)
Taking the ratio of vplanet to vearth, we get:
vplanet/vearth = √(24GπR^2) / √(8GπR^2)
vplanet/vearth = √(24/8)
vplanet/vearth = √3
Therefore, the ratio of the escape speeds from the surface of the planet to the surface of the Earth is 3:1, which corresponds to option D.
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Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer?
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Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer?.
Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer? for NEET 2024 is part of NEET preparation. The Question and answers have been prepared according to the NEET exam syllabus. Information about Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for NEET 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer?.
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Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer?, a detailed solution for Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of Radius of a planet is three times the radius of the earth and it has same average density as that of the earth. The ratio of escape speeds from surface of the planet and that from the surface of earth isa)8 : 1b)6 : 1c)7 : 1d)3 : 1Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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