Test: Gravitational potential energy & Escape speed (7 August) - JEE MCQ

Escape velocity

• It is the minimum velocity with which a body should be projected from the surface of the planet so that the body reaches infinity.
• It is given as,
  

Where G = gravitational constant, M = mass of the planet, and R = radius of the planet

Calculations: 
As we know that the escape velocity from the surface of the earth is given as,

Where G = gravitational constant, M = mass of the earth, and R = radius of the earth

• By equation 1 it is clear that the escape velocity depends upon the mass and the radius of the earth but it is independent of the mass, shape, and size of the projected body.
• So the escape velocity for stone A and stone B will be equal.
• Therefore the ratio of the escape velocity of stone A to stone B from the surface of the earth is 1. Hence, option 3 is correct.

Concept:​​​

• Gravitational Potential Energy: It is the energy possessed by a body at a certain point when work is done by the force of gravity in bringing the object from infinity to that point.
• The gravitational potential energy between two masses m₁ and m₂ separated by a distance r is given by 

Explanation:

• The gravitational potential (V) is the gravitational potential energy per unit mass. 
• 
• ​At an infinite distance, the gravitational field is assumed to be zero and hence, it will not interact with the mass with regards to gravitational potential. Thus, when E = 0, V = 0.
• We know that the gravitational field inside a spherical shell is zero. The gravitational potential inside a shell is the same as that on the shell 
  
• V ≠ 0, E ≠ 0 is a general case when there exists a gravitational potential in a gravitational field.
• Hence, all the given options are correct.

Escape Velocity (v): 

• The minimum velocity required to escape the gravity of the earth is called escape velocity.
• If an object is through with this speed or above, it will escape the gravity of earth and will not return back. 
• Escape velocity (v_e) for a body on earth is given as 
  
  G is the universal gravitational constant, M is the mass of earth, R is the radius of the earth.

Explanation:

The value of the Universal gravitational constant is G = 6.67 × 10^-11 N kg^-2 m².

Mass of Earth is  M = 5.972 x 10²⁴ kg

Radius of earth is R = 6370000 m

Putting this in the expression of escape velocity we will get the value of it around 11.2 km/s. 

So the correct option is 11.2 km/s

Gravitational force exists between each and every object, but it cannot be felt unless the mass of the objects is very high, such as in planets.

Concept:

• The escape velocity is the lowest velocity that a body must have in order to escape the gravitational attraction of a particular planet.

• Where, G = gravitational constant, R = radius of earth, M = mass of earth
• For earth escape velocity is 11.2 km/s

Calculation:

Let the mass of the earth is M, then the planet mass M' = 1000M,

Radius is R, then R' = 10R

The escape velocity of the earth,

Now, the Escape velocity of the planet is

The gravitational potential energy at any point can be defined as work done in bringing a body of mass m from infinite distance to its present location

Hence gravitational potential energy of mass M and m at some distance r can be expressed as

Here r = R+h, if the satellite is at some height h from the surface and R, is the radius of the planet

Hence gravitational potential energy of mass M and m at some distance R on the surface of the planet

Here,

r = Distance between satellite from the centre of the planet

h = Distance between satellite from the centre of the planet

M = mass of the planet

m = mass of the satellite

G = gravitational constant
Explanation:

From the above explanation, we can see that gravitational potential at some hight from the earth's centre (r) can be expressed as 

From the above explanation, we can see that gravitational potential at some hight from the earth's centre (r) can be expressed as 

Escape Velocity (v_e): 

The minimum velocity with which a body must be projected up to enable it to just overcome the gravitational pull is known as escape velocity.

Orbital Velocity (v_o):

Satellites are natural or artificial bodies describing an orbit around a planet under its gravitational attraction.

• The orbital velocity of a satellite is the velocity required to put the satellite into its orbit around the earth.
• For the revolution of a satellite around the earth, the gravitational pull provides the required centripetal force.

Explanation:

• This work required to project the body to escape the gravitational pull is performed on the body by providing an equal amount of kinetic energy to it at the surface of the earth.

If v_e is the required escape velocity, then kinetic energy which should be given to the body to escape the gravitational pull is

Where, G = universal gravitational constant, M = mass of the earth, R = radius of the earth and v_e = escape velocity of the body.

For the revolution of a satellite around the earth, the gravitational pull provides the required centripetal force.

where, G = universal gravitational constant, M = mass of the earth, r = distance between the earth and the satellite and v_o = orbital velocity of the satellite.
Divide equations (1) and (2), we get

Concept:

Gravitational potential energy is the energy stored in any two objects due to their gravity and the distance between them.

Where, m is the mass of the first body, M = mass of the second body, R is the distance between them, G = the universal gravitational constant,

Calculation:

Let the mass m is at distance r from one of the spheres and the mass of spheres is M

Then the total potential energy of the system is,

Now at  the potential energy of the system is,

Now at  the potential energy of the system is,

now at r = 3R/4,  the potential energy of the system is,

Now from equations 1, 2, and 3,

U₁ is less than U₂ and U₂ is greater than U₃, so the potential energy first increases and then decreases. 

Concept: 

• Escape velocity: The minimum velocity required to escape the gravitational field of the earth is called escape velocity. It is denoted by Ve.

The escape velocity on earth is given by:

Where G is the universal gravitational constant, M is the mass of the earth and R is the radius of the earth, m is the mass of the body and h is the height above the earth’s surface
 

Explanation:

According to the formula of the escape velocity of a body,

• It is independent of the mass of the body. So option 4 is correct.

Comet:

Comets are relatively small and icy celestial bodies revolving round the sun. When a comet comes near the sun, some of the 'ice' in it turns into gas. The gas and loose dust, free from ice, create a long illuminating tail that streams behind the comet. 

Asteroid:

These are very small planets or planetoids that are found mainly in a belt between the orbit of Mars and Jupiter. Each asteroid has its own orbit and the orbits of all of them are spread over a large distance, forming a band.

Meteors:

Meteors are pieces of stony or metallic rock widely scattered throughout the solar system. They travel at high speeds around the sun. When meteors come close enough to the earth, they are pulled towards the earth by its gravitational force. As the meteor passes through the earth's atmosphere, it gets burnt due to the immense heat produced by friction with the atmosphere, leaving a brilliant trail of light behind it. Such a meteor is also called a shooting star.