Question

If an object is projected vertically upwards with speed,half the escape speed of earth, then the maximum height attained by it is [R is radius of earth]

Answer 1

We know that the escape velocity of Earth = Ve = √[2gR]

g = gravity due to Earth

1/2 m (Ve)² - G Mm/R = 0 for the object to escape from Earth.

(or, KE + PE = 0)

When the body is projected from Earth's surface:

Given u = initial velocity = Ve /2 = √(gR/2)

Initial KE = 1/2 m u² = m g R/4 = G M m /(4R)

Initial PE = - GMm/R

Total initial energy = - 3 GMm /(4R)

When the body reaches a height h above surface of Earth and stops:

v = 0. Final KE = 0.

Final PE = - G M m/ (R+h)

From conservation of Energy:

- GMm/(R+h) = - 3Gm/ (4R)

3 (R+h) = 4 R

h = R/3

Answer is R/3.