body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If R is the radius of the earth, maximum height attained by the body from the surface of the earth is

Question

body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If R is the radius of the earth, maximum height attained by the body from the surface of the earth is (A) R/6 (B) R/3 (C) 2R/3 (D) R

Tardigrade Admin · Accepted Answer

Applying conservation of mechanical energy, (1/2) m((ve/2))2=(m g h/1+(h)R) or (ve2/8)=(g h/1+(h)R) Ð¾Ð³ (2 g R/8)=(g h/1+h / R) ldots (i) (∵ ve=√2 g R) On solving Eq. (i), we get h=(R/3).

Q. body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If $R$ is the radius of the earth, maximum height attained by the body from the surface of the earth is

R/6

R/3

2R/3

R

Applying conservation of mechanical energy,
$\frac{1}{2} m (\frac{v _{e}}{2})^{2} = \frac{m g h}{1 + \frac{h}{R}}$
or $\frac{v _{e}^{2}}{8} = \frac{g h}{1 + \frac{h}{R}}$
ог $\frac{2 g R}{8} = \frac{g h}{1 + h / R} \dots$ (i)
$(∵ v_{e} = 2 g R)$
On solving Eq. (i), we get $h = \frac{R}{3}$ .

Q. body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If R is the radius of the earth, maximum height attained by the body from the surface of the earth is

R/6

R/3

2R/3

R

Applying conservation of mechanical energy, 21​m(2ve​​)2=1+Rh​mgh​ or 8ve2​​=1+Rh​gh​ ог 82gR​=1+h/Rgh​… (i) (∵ve​=2gR​) On solving Eq. (i), we get h=3R​.

Q. body is projected vertically upwards from the surface of the earth with a velocity equal to half the escape velocity. If $R$ is the radius of the earth, maximum height attained by the body from the surface of the earth is

Applying conservation of mechanical energy,
$\frac{1}{2} m (\frac{v _{e}}{2})^{2} = \frac{m g h}{1 + \frac{h}{R}}$
or $\frac{v _{e}^{2}}{8} = \frac{g h}{1 + \frac{h}{R}}$
ог $\frac{2 g R}{8} = \frac{g h}{1 + h / R} \dots$ (i)
$(∵ v_{e} = 2 g R)$
On solving Eq. (i), we get $h = \frac{R}{3}$ .