Qno.22-What do you mean by escape velocity and find expression for escape velocity from earth surface? Qno 23-What do you mean by angular momentum and establish a relation between angular momentum and angular velocity? Q

Step-by-step solution

1. 1

   Problem Statement

   Define escape velocity and derive its expression from Earth's surface.
2. 2

   Escape Velocity Definition

   Escape velocity is the minimum velocity required for an object to escape the gravitational field of a celestial body.
3. 3

   Energy Conservation

   Apply the principle of conservation of mechanical energy.

   $$E_{initial} = E_{final}$$
4. 4

   Initial Energy

   Initial kinetic energy and potential energy at Earth's surface.

   $$E_{initial} = \frac{1}{2}mv_e^2 - \frac{GMm}{R_e}$$
5. 5

   Final Energy

   Final kinetic and potential energy at infinity (escaped).

   $$E_{final} = 0 + 0 = 0$$
6. 6

   Equate Energies

   Equating initial and final energies.

   $$\frac{1}{2}mv_e^2 - \frac{GMm}{R_e} = 0$$
7. 7

   Solve for $v_e$

   Rearrange the equation to find $v_e$.

   $$v_e = \sqrt{\frac{2GM}{R_e}}$$
8. 8

   Alternative Form

   Using $g = \frac{GM}{R_e^2}$, substitute $GM = gR_e^2$.

   $$v_e = \sqrt{2gR_e}$$
9. 9

   Final Expression

   The expression for escape velocity from Earth's surface.

   $$v_e = \sqrt{\frac{2GM}{R_e}} = \sqrt{2gR_e}$$