Find the best tutors and institutes for Class 11 Tuition
Q8.18:
The escape speed of a projectile on the earth’s surface is 11.2 km s$^{–1}$. A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets.
Solution :
Step 1: Establishing the Principle of Conservation of Mechanical Energy
According to the Law of Conservation of Energy, the total mechanical energy of an object at the Earth's surface must equal the total mechanical energy when the object is at an infinite distance ($r \to \infty$) from the Earth [Per the Work-Energy Theorem].
The total energy $E$ at the surface is given by:
$E_i = K_i + U_i = \frac{1}{2}mv^2 - \frac{GMm}{R}$
Where:
- $v$ = Initial velocity of projection
- $G$ = Universal gravitational constant
- $M$ = Mass of the Earth
- $m$ = Mass of the projectile
- $R$ = Radius of the Earth
Step 2: Defining Escape Velocity
The escape velocity ($v_e$) is defined as the minimum velocity required for an object to reach infinity with zero kinetic energy. Thus, $\frac{1}{2}mv_e^2 = \frac{GMm}{R}$, which simplifies to $v_e = \sqrt{\frac{2GM}{R}}$.
Given $v_e = 11.2 \text{ km s}^{-1}$, we have the relation $\frac{GM}{R} = \frac{v_e^2}{2}$.
Step 3: Calculating Energy at Infinite Distance
At an infinite distance, the potential energy $U_f$ becomes zero. Let $v_f$ be the final speed of the body. The final kinetic energy is $\frac{1}{2}mv_f^2$.
$E_f = \frac{1}{2}mv_f^2 + 0$
Equating $E_i = E_f$:
$\frac{1}{2}mv^2 - \frac{GMm}{R} = \frac{1}{2}mv_f^2$
Step 4: Algebraic Substitution
Given $v = 3v_e$, substitute this into the equation:
$\frac{1}{2}m(3v_e)^2 - \frac{GMm}{R} = \frac{1}{2}mv_f^2$
Divide throughout by $\frac{1}{2}m$:
$(3v_e)^2 - \frac{2GM}{R} = v_f^2$
Since $\frac{2GM}{R} = v_e^2$ [Derived from Step 2]:
$9v_e^2 - v_e^2 = v_f^2$
$v_f^2 = 8v_e^2$
$v_f = \sqrt{8}v_e = 2\sqrt{2}v_e$
Step 5: Numerical Computation
Substitute $v_e = 11.2 \text{ km s}^{-1}$:
$v_f = 2 \times 1.414 \times 11.2 \text{ km s}^{-1}$
$v_f = 2.828 \times 11.2 \text{ km s}^{-1}$
$v_f \approx 31.68 \text{ km s}^{-1}$
Final Solution: The speed of the body far away from the earth is $31.68 \text{ km s}^{-1}$.
More Questions from Class 11 Physics Gravitation EXERCISES
- Q7.1(a): Answer the following : You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means ?
- Q7.1(b): Answer the following : An astronaut inside a small space ship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity ?
- Q7.1(c): Answer the following : If you compare the gravitational force on the earth due to the sun to that due to the moon, you would find that the Sun’s pull is greater than the moon’s pull. (you can check this yourself using the data available in the succeeding exercises). However, the tidal effect of the moon’s pull is greater than the tidal effect of sun. Why ?
- Q7.10: Choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 7.11)
- Q7.11: For the above problem, the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow
- Q7.12: A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero ? Mass of the sun = (2\times10^{30}) kg, mass of the earth = (6\times10^{24}) kg. Neglect the effect of other planets etc. (orbital radius = (1.5 \times 10^{11}) m).
- Q7.13: How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is (1.5 \times 10^{8}) km.
- Q7.14: A saturn year is 29.5 times the earth year. How far is the saturn from the sun if the earth is (1.50 \times 10^{8}) km away from the sun ?
- Q7.15: A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth ?
- Q7.16: Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface ?
- Q7.17: A rocket is fired vertically with a speed of (5 \text{ km s}^{-1}) from the earth’s surface. How far from the earth does the rocket go before returning to the earth ? Mass of the earth = (6.0 \times 10^{24}) kg; mean radius of the earth = (6.4 \times 10^{6}) m; G = (6.67 \times 10^{-11} \text{ N m}^{2} \text{ kg}^{-2}).
- Q7.18: The escape speed of a projectile on the earth’s surface is (11.2 \text{ km s}^{-1}). A body is projected out with thrice this speed. What is the speed of the body far away from the earth? Ignore the presence of the sun and other planets.
- Q7.19: A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = (6.0\times10^{24}) kg; radius of the earth = (6.4 \times 10^{6}) m; G = (6.67 \times 10^{-11} \text{ N m}^{2} \text{ kg}^{-2}).
- Q7.2(a): Choose the correct alternative : Acceleration due to gravity increases/decreases with increasing altitude.
- Q7.2(b): Choose the correct alternative : Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).
- Q7.2(c): Choose the correct alternative : Acceleration due to gravity is independent of mass of the earth/mass of the body.
- Q7.2(d): Choose the correct alternative : The formula (–G M m (\frac{1}{r_{2}} – \frac{1}{r_{1}})) is more/less accurate than the formula (m g (r_{2} – r_{1})) for the difference of potential energy between two points (r_{2}) and (r_{1}) distance away from the centre of the earth.
- Q7.20: Two stars each of one solar mass (= (2\times10^{30}) kg) are approaching each other for a head on collision. When they are a distance (10^{9}) km, their speeds are negligible. What is the speed with which they collide ? The radius of each star is (10^{4}) km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
- Q7.21: Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational force and potential at the mid point of the line joining the centres of the spheres ? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable ?
- Q7.3: Suppose there existed a planet that went around the Sun twice as fast as the earth. What would be its orbital size as compared to that of the earth ?
- Q7.4: Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is (4.22 \times 10^{8}) m. Show that the mass of Jupiter is about one-thousandth that of the sun.
- Q7.5: Let us assume that our galaxy consists of (2.5 \times 10^{11}) stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution ? Take the diameter of the Milky Way to be (10^{5}) ly.
- Q7.6(a): Choose the correct alternative: If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
- Q7.6(b): Choose the correct alternative: The energy required to launch an orbiting satellite out of earth’s gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earth’s influence.
- Q7.7(a): Does the escape speed of a body from the earth depend on the mass of the body,
- Q7.7(b): Does the escape speed of a body from the earth depend on the location from where it is projected,
- Q7.7(c): Does the escape speed of a body from the earth depend on the direction of projection,
- Q7.7(d): Does the escape speed of a body from the earth depend on the height of the location from where the body is launched?
- Q7.8(a): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant linear speed,
- Q7.8(b): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant angular speed,
- Q7.8(c): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant angular momentum,
- Q7.8(d): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant kinetic energy,
- Q7.8(e): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant potential energy,
- Q7.8(f): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q7.9: Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet, (b) swollen face, (c) headache, (d) orientational problem.
- Q8.1(a): Answer the following : (a) You can shield a charge from electrical forces by putting it inside a hollow conductor. Can you shield a body from the gravitational influence of nearby matter by putting it inside a hollow sphere or by some other means ?
- Q8.1(b): Answer the following : (b) An astronaut inside a small space ship orbiting around the earth cannot detect gravity. If the space station orbiting around the earth has a large size, can he hope to detect gravity ?
- Q8.1(c): Answer the following : (c) If you compare the gravitational force on the earth due to the sun to that due to the moon, you would find that the Sun’s pull is greater than the moon’s pull. (you can check this yourself using the data available in the succeeding exercises). However, the tidal effect of the moon’s pull is greater than the tidal effect of sun. Why ?
- Q8.10: Choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12)
- Q8.11: Choose the correct answer from among the given ones: The gravitational intensity at the centre of a hemispherical shell of uniform mass density has the direction indicated by the arrow (see Fig 8.12), the direction of the gravitational intensity at an arbitrary point P is indicated by the arrow.
- Q8.12: A rocket is fired from the earth towards the sun. At what distance from the earth’s centre is the gravitational force on the rocket zero ? Mass of the sun = $2 \times 10^{30}$ kg, mass of the earth = $6 \times 10^{24}$ kg. Neglect the effect of other planets etc. (orbital radius = $1.5 \times 10^{11}$ m).
- Q8.13: How will you ‘weigh the sun’, that is estimate its mass? The mean orbital radius of the earth around the sun is $1.5 \times 10^8$ km.
- Q8.14: A saturn year is 29.5 times the earth year. How far is the saturn from the sun if the earth is $1.50 \times 10^8$ km away from the sun ?
- Q8.15: A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height equal to half the radius of the earth ?
- Q8.16: Assuming the earth to be a sphere of uniform mass density, how much would a body weigh half way down to the centre of the earth if it weighed 250 N on the surface ?
- Q8.17: A rocket is fired vertically with a speed of 5 km s$^{-1}$ from the earth’s surface. How far from the earth does the rocket go before returning to the earth ? Mass of the earth = $6.0 \times 10^{24}$ kg; mean radius of the earth = $6.4 \times 10^6$ m; G = $6.67 \times 10^{–11}$ N m$^2$ kg$^{–2}$.
- Q8.19: A satellite orbits the earth at a height of 400 km above the surface. How much energy must be expended to rocket the satellite out of the earth’s gravitational influence? Mass of the satellite = 200 kg; mass of the earth = $6.0 \times 10^{24}$ kg; radius of the earth = $6.4 \times 10^6$ m; G = $6.67 \times 10^{–11}$ N m$^2$ kg$^{–2}$.
- Q8.2(a): Choose the correct alternative : (a) Acceleration due to gravity increases/decreases with increasing altitude.
- Q8.2(b): Choose the correct alternative : (b) Acceleration due to gravity increases/decreases with increasing depth (assume the earth to be a sphere of uniform density).
- Q8.2(c): Choose the correct alternative : (c) Acceleration due to gravity is independent of mass of the earth/mass of the body.
- Q8.2(d): Choose the correct alternative : (d) The formula $-G Mm(\frac{1}{r_2} – \frac{1}{r_1})$ is more/less accurate than the formula $mg(r_2 – r_1)$ for the difference of potential energy between two points $r_2$ and $r_1$ distance away from the centre of the earth.
- Q8.20: Two stars each of one solar mass (= $2 \times 10^{30}$ kg) are approaching each other for a head on collision. When they are a distance $10^9$ km, their speeds are negligible. What is the speed with which they collide ? The radius of each star is $10^4$ km. Assume the stars to remain undistorted until they collide. (Use the known value of G).
- Q8.21: Two heavy spheres each of mass 100 kg and radius 0.10 m are placed 1.0 m apart on a horizontal table. What is the gravitational force and potential at the mid point of the line joining the centres of the spheres ? Is an object placed at that point in equilibrium? If so, is the equilibrium stable or unstable ?
- Q8.3: Suppose there existed a planet that went around the sun twice as fast as the earth. What would be its orbital size as compared to that of the earth ?
- Q8.4: Io, one of the satellites of Jupiter, has an orbital period of 1.769 days and the radius of the orbit is $4.22 \times 10^8$ m. Show that the mass of Jupiter is about one-thousandth that of the sun.
- Q8.5: Let us assume that our galaxy consists of $2.5 \times 10^{11}$ stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution ? Take the diameter of the Milky Way to be $10^5$ ly.
- Q8.6(a): Choose the correct alternative: (a) If the zero of potential energy is at infinity, the total energy of an orbiting satellite is negative of its kinetic/potential energy.
- Q8.6(b): Choose the correct alternative: (b) The energy required to launch an orbiting satellite out of earth’s gravitational influence is more/less than the energy required to project a stationary object at the same height (as the satellite) out of earth’s influence.
- Q8.7(a): Does the escape speed of a body from the earth depend on (a) the mass of the body,
- Q8.7(b): Does the escape speed of a body from the earth depend on (b) the location from where it is projected,
- Q8.7(c): Does the escape speed of a body from the earth depend on (c) the direction of projection,
- Q8.7(d): Does the escape speed of a body from the earth depend on (d) the height of the location from where the body is launched?
- Q8.8(a): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (a) linear speed, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(b): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (b) angular speed, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(c): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (c) angular momentum, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(d): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (d) kinetic energy, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(e): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (e) potential energy, throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.8(f): A comet orbits the sun in a highly elliptical orbit. Does the comet have a constant (f) total energy throughout its orbit? Neglect any mass loss of the comet when it comes very close to the Sun.
- Q8.9(a): Which of the following symptoms is likely to afflict an astronaut in space (a) swollen feet,
- Q8.9(b): Which of the following symptoms is likely to afflict an astronaut in space (b) swollen face,
- Q8.9(c): Which of the following symptoms is likely to afflict an astronaut in space (c) headache,
- Q8.9(d): Which of the following symptoms is likely to afflict an astronaut in space (d) orientational problem.
CBSE Solutions for Class 11 Physics Gravitation
Chapters in CBSE - Class 11 Physics
Top Tutors who teach Gravitation
Super Tutor
Banjara Hills, Hyderabad
12 yrs of
Exp
74 students
1638 hrs of classes completed
800
per hour
Online Classes
Tutor's home
Vinay Raj Katakam is a passionate educator with over 10 years of experience teaching Accountancy and Economics. He holds an M.Com degree from Osmania University, where he graduated as a topper, along with CA-IPCC and NET qualifications. Dedicated to his students’ success, Vinay goes above and beyond to ensure their doubts are thoroughly clarified and their understanding of concepts is complete. His contributions to education have earned him 2 international and 2 national awards, and his research has been published in reputed journals. He is also recognized as a leading guest lecturer at various universities across India. Vinay’s teaching journey began during his first year of college, driven by his passion for sharing knowledge. Before the pandemic, he successfully ran his own educational institution, mentoring numerous students toward academic excellence. Currently, he teaches online through platforms like UrbanPro, specializing in ISC, CBSE (11th and 12th), and IGCSE (10th, 11th, and 12th) students. Known for his ability to simplify complex topics, develop analytical skills, and provide personalized support, Vinay uses digital tools to create interactive lessons with assignments and questionnaires. All his students consistently score above 90%, with many achieving perfect marks. While stock market trading is his profession, teaching remains his true passion.
I was initially hesitant to join tuition due to certain unique circumstances, but learning with Vinay Raj Katakam has been a very ideal experience. He is patient, understanding, and highly receptive to his students’ perspectives, which makes his teaching especially effective. His explanations are clear, coherent, and easy to grasp, allowing complex concepts to feel approachable. Overall, studying under his guidance has been both insightful and rewarding.
Super Tutor
Arrah Chowk, Arrah
14 yrs of
Exp
27 students
794 hrs of classes completed
300
per hour
Online Classes
Student's Home
I am an MSc Chemistry tutor with 14 years of teaching experience in online and offline classes. I teach: • Class 11 Chemistry • Class 12 Chemistry • CBSE / ISC / State Board • NEET Chemistry • JEE Chemistry My teaching method focuses on: • Strong concept building • Easy explanation of difficult topics • Numericals practice • Regular tests and assignments • NCERT based preparation • Doubt solving sessions I help students improve marks, confidence, and understanding in Chemistry. Languages: Hindi & English 93+ positive reviews and 100+ students taught. Book a free demo class and discuss your learning goals.
I have had a pleasant experience doing these chemistry classes. The concepts were taught nicely and classes were very interactive as well.
Top Tutor
Ashok Nagar D' Souza Layout, Bangalore
10 yrs of
Exp
4 students
1045 hrs of classes completed
700
per hour
Online Classes
10+ years of experience teaching Mathematics for grades 10 to 12. • Consistent record of 100% academic results. • Very friendly and supportive learning environment. • Encourages students to ask questions freely and confidently. • Strong focus on conceptual clarity and problem-solving skills. • Helps students build confidence and excel in Mathematics. • Fee structure for one-to-one classes: ₹800 per hour.
He is a very good and experienced teacher who helps he improve at maths significantly. I used to get really bad marks earlier and then I started doing classes with him and I saw a significant improvement in my marks. He is one of the best maths teachers I have ever had.
Super Tutor
Nerul Sector 20, Thane
10 yrs of
Exp
72 students
5982 hrs of classes completed
1200
per hour
Online Classes
I have done b.Tech in electrical engineering from IIT KANPUR. I taught one year in akash institute patna, 2 years in fiitjee patna. Currently I am teaching in unacademy. I have taught more than 6000 jee and neet aspirants. In which more than 500 students are selected in iitand aiims. If some one is interested to improve their physics upto jee advanced level take help with me .And see the magic. I can help students in their all type of assignment from any institute like allen, fiitjee, narayna, akash, resonance. I can help students to solve both volume of h.C verma within two month if some one is intrested in solving I.E.Irodov join me and see the next level of physics My way of teaching is so easy and so advanced that student will realyy enjoy it. Don't waste your time in making decision just join and experience my level
My Physics teacher explains concepts very clearly and makes even difficult topics easier to understand. The classes are engaging, and the examples used help connect the concepts to real-life situations. I also appreciate the way doubts are addressed patiently and without making students feel uncomfortable about asking questions. The teacher encourages us to think logically rather than just memories formulas, which has helped me develop a better understanding of Physics. Overall, the teaching is effective, supportive, and motivating.
Top Tutor
Sakinaka East, Mumbai
5 yrs of
Exp
47 hrs of classes completed
1500
per hour
Online Classes
Student's Home
Tutor's home
Starting from fundamentals of the subject and explaining in such a way that the student feels comfortable with the subject. Repeating the difficult topics and conducting weekly tests so that the student gets the confident over the subject.
