Question

A particle of mass m is thrown upwards from the
surface of the earth, with a velocity u. The mass and
the radius of the earth are, respectively, M and R.
G is gravitational constant and g is acceleration due
to gravity on the surface of the earth. The minimum
value of u so that the particle does not return back
to earth, is?​

Answer 1

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✤ Required Answer:

✒ GiveN:

• A particle of mass = m is thrown upwards from the surface of the earth, with a velocity = u.

• The mass and the radius of the earth are = M & R.

• G is gravitational constant and g is acceleration due to gravity on the surface of the earth.

✒ To FinD:

• The minimum value of u so that the particle does not return back to earth .

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✤ How to solve ?

In the question we have to find The minimum value of u so that the particle does not return back to earth and it will happen when the energy of particle will be infinite which is equal to zero (0) .

[ See the attachment ]

Now let's solve it !

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✤ Solution :

↪ Potential energy of a body of mass m are the surface of the earth

★ Energy = ( ( potential energy × mass of earth × mass of particle ) ÷ radius of earth ) + kinetic energy

Now let's write the value

• Energy = E
• Potential Energy = G
• Mass of earth = M
• Mass of particle = m
• Radius of Earth = R
• Kinetic energy = ½mu²

$: \Longrightarrow{{\sf{{E = \dfrac{ - GMm}{R} + \dfrac{1}{2} mu ^{2} }}}}$

Now we know that in order to find the minimum value of u so that the particle does not return back to earth and it will happen when the energy of particle will be infinite which is equal to zero (0)

Hence ,

$: \Longrightarrow{{\sf{{0 = \dfrac{ - GMm}{R} + \dfrac{1}{2} mu ^{2} }}}}$

$: \Longrightarrow{{\sf{{u ^{2} = \dfrac{ 2 GM}{R} }}}}$

$: \Longrightarrow{{\sf{{u = \blue{ \sqrt{\dfrac{ 2 GM}{R}} }}}}}$

$: \Longrightarrow{{\sf {u = \pink{\sqrt{ 2 gR}}..( \because g = \dfrac{GM}{ {R}^{2} }) }}}$

☀️ Hence, solved !!

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Answer 2

Question -

A particle of mass m is thrown upwards from the surface of the earth, with a velocity u. The mass and the radius of the earth are, respectively, M and R. G is gravitational constant and g is acceleration due to gravity on the surface of the earth. The minimum value of u so that the particle does not return back to earth, is?

Answer -

u = √2GM/R