Question

If a body is projected with a speed less than the escape velocity

Answer 1

Explanation:

If a body is projected with a speed less than the escape velocity then the body will perform projectile motion and land on earth surface.

Answer 2

Hey Nitish! Thank you for putting up the doubt.

Given :-

• body is projected with a speed less than the escape speed

To find :-

• To derive the equation for the height till which the projectile reaches.

Solution :-

Consider :

• mass of the body from where it is projected = M
• mass of the projected body = m
• radius of the the planet = R
• the height the body attains with the needed speed = h
• the needed speed at the attained height = v
• let the escape velocity = V

Now, through the concepts of conservation of mechanical energy for the given case, we have:

(K.E. + u) on earth surface = (K.E. + u) at height

1/2 mv² - GMm/R = 0 - GMm/R + h

m( v²/2 - GM/R ) = m( - GM/R + h)

v²/2 - GM/R = - GM/ R + h

v²/2 = GM/R - GM/R + h

v²/2 = GM ( 1/ R - 1/R + h)

v² = 2 GM [ R + h - R/ R(R + h) ]

v² = 2 GM [ h/ R ( R + h)]

v² = 2 GM [1/R ( h/ R + h)]

v² = 2 GM/ R [ h/ R + h]

v² = V [ h / R + h ]

v²/ V = h / R + h

v²/ V = h/R + h/h

v²/ V = h/ R + 1

(v²/ V - 1) = h/R

h = R/ (v²/ v- 1)

Hence, the height till which the projectile reaches when projected with a speed less than the escape speed is h = R/ (v²/ v- 1).