Show that the escape speed n for a body located on the surface of a planet of radius r and mass M is given by the expression v = sqrt( (2* G * M)/r)
now im using (K+Ug)i = (K+Ug)k
where Kf = 0 because final velocity is zero
and Ugf = 0 because its final distance is infinity
To show that the escape speed for a body located on the surface of a planet of radius r and mass M is given by the expression v = sqrt((2 * G * M) / r), you can use the principle of conservation of mechanical energy.
1. Start by considering the initial state of the body when it is on the surface of the planet. The body has both kinetic energy (K) and gravitational potential energy (Ug).
2. Write the equation for the total mechanical energy (K + Ug) of the body in the initial state as (K + Ug)i.
3. Now, consider the final state of the body when it has escaped the gravitational pull of the planet. At this point, the body is at an infinite distance from the planet, so its final distance is considered to be infinity.
4. The final velocity of the body is zero (vf = 0) because it has just escaped the gravitational pull.
5. Write the equation for the total mechanical energy (K + Ug) of the body in the final state as (K + Ug)f.
6. Apply the principle of conservation of mechanical energy, which states that the total mechanical energy of a system remains constant if no external forces are acting on it.
7. Therefore, equate the initial mechanical energy (K + Ug)i to the final mechanical energy (K + Ug)f and solve for v, the escape speed.
(K + Ug)i = (K + Ug)f
1/2 * m * vi^2 - (G * M * m) / r = 0 + 0
1/2 * m * vi^2 - G * M * m / r = 0
8. Rearrange the equation to solve for vi (initial velocity of the body).
1/2 * m * vi^2 = G * M * m / r
vi^2 = 2 * G * M / r
9. Take the square root of both sides to solve for vi.
vi = sqrt(2 * G * M / r)
10. The initial velocity (vi) represents the escape speed (v), as it is the velocity required for the body to escape the gravitational pull of the planet.
Therefore, the escape speed (v) for a body on the surface of a planet of radius r and mass M is given by the expression v = sqrt((2 * G * M) / r), where G is the gravitational constant.