Gravitation

Escape Velocity

Suppose you are playing cricket and you hit the ball with some velocity, the ball will again come down on the surface of the ground. But in case you hit the ball with greater velocity, the ball will escape out of the gravitational field. This is what we call escape velocity. Let us learn more about this.

Suggested Videos

Play
Play
Play
previous arrow
next arrow
previous arrownext arrow
Slider

 

Escape Velocity

To understand the term Escape Velocity let us carry out a small activity.

Escape Velocity

                                                                                                                     ( Source: The cheap route )

Suppose you are having a ball in your hand. You throw a ball in the air and you see that it comes down. We know that it comes back because of the force of gravitation. Now you throw the ball with greater velocity, in this case, the balls reach a greater height but eventually, it comes down and falls on the surface of the ground.

Because it still experiences the force of attraction by the surface of the earth. Now suppose you throw the ball with such a high velocity that it never comes back on the ground. This is where escape velocity comes into the picture. Escape velocity is the velocity that a body must attain to escape a gravitational field.

So if you throw the ball with the velocity which is at least equal to the escape velocity, in that case, the ball will go out of the gravitational field.

Mathematical Expression

Suppose the ball is initially in your hand. So that is the initial position of the ball. Now throw the ball at a greater velocity, that it never comes back. As we don’t know where did the ball go, so its final velocity is ∞. So with this assumption let us derive the expression.

At initial position,

Total energy = Kinetic energy + Potential energy

Or, T.E = K.E + P.E

Kinetic Energy = \( \frac{1}{2} \) mv²

Potential Energy = \( \frac{- GMm}{R_e + h}\)

Here M is the mass of the earth and m is the mass of the ball.  At the earth’s surface, P.E (0) = 0.

Hence, T. E (0) =  \( \frac{1}{2} \) mv²

At final position (∞),

K.E = \( \frac{1}{2} \) mvf²

P.E =  \( \frac{-GM_em}{R_e + h} \) = 0 {h =∞}

Now by the law of conservation of energy, the total energy at the initial position should be equal to the final position.

T.E (∞)  = T.E (0)

Or, \( \frac{1}{2} \) mvf² = \( \frac{1}{2} \) m vi² –  \( \frac{GM_em}{R_e + h} \)

L.H.S has to be alawys +ve, which implies

\( \frac{1}{2} \) m vi² – \( \frac{GM_em}{R_e + h} \)  ≥ 0

⇒ \( \frac{1}{2} \) m vi² = \( \frac{GM_em}{R_e + h} \)

⇒ vi² = \( \frac{2GM_e}{R_e + h} \)

Assume the ball is thrown from earth’s surface.

h << Re ⇒Re+h ~ Re

⇒  vi² = \( \frac{2GM_e}{R_e } \)

⇒ vi =  \( \sqrt{\frac{2GM_e}{R_e }} \)

This is the velocity in which the objects never comes back. In terms of ‘ g ‘

g = \( \frac{GM}{R²} \)

⇒ gRe =  \( \frac{GM_e}{R_e} \)

ve  = √(2gRe)

Solved Questions For You

Q1. The escape velocity of a particle depends on its mass m as:

  1. mass m as m²
  2. as m-1
  3. mass m as m0
  4. as m1

Ans: C. Escape velocity, ve= √2gR. It is independent of the mass of the particle. Thus, it will depend on m0

Q2. The earth retains its atmosphere. This is due to:

  1. the special shape of the earth
  2. the escape velocity which is greater than the mean speed of the atmospheric molecules.
  3. the escape velocity which is less than the mean speed of the atmospheric molecules.
  4. the suns gravitational effect.

Ans: B. The earth retains its atmosphere. This is due to the escape velocity is been greater than the mean speed of the atmospheric molecules.

Share with friends

Customize your course in 30 seconds

Which class are you in?
5th
6th
7th
8th
9th
10th
11th
12th
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
tutor
tutor
Ashhar Firdausi
IIT Roorkee
Biology
tutor
tutor
Dr. Nazma Shaik
VTU
Chemistry
tutor
tutor
Gaurav Tiwari
APJAKTU
Physics
Get Started

2 responses to “Kepler’s Law of Planetary Motions – Orbits, Areas, Periods”

  1. Sahil says:

    When earth is near know it move faster some gravity of earth act on it and it produce restriction so speed may be slownear sun

  2. Sahil says:

    When earth is near the sun how it move faster some gravity of sun act on it and it produce restriction and speed may be slow down

Leave a Reply

Your email address will not be published. Required fields are marked *

Download the App

Watch lectures, practise questions and take tests on the go.

Customize your course in 30 seconds

No thanks.