A projectile is projected with velocity k v_e in vertically upwar… - Vidyadip

MCQ

A projectile is projected with velocity $k v_e$ in vertically upward direction from the ground into the space. ( $v_e$ is escape velocity and $k<1$ ). If air resistance is considered to be negligible then the maximum height from the centre of earth to which it can go, will be $:(R=$ radius of earth $)$

A
$\frac{R}{k^2+1}$
B
$\frac{R}{k^2-1}$
✓
$\frac{R}{1-k^2}$
D
$\frac{R}{k+1}$

✓

Answer

Correct option: C.

$\frac{R}{1-k^2}$

Kinetic energy $=$ Potential energy
$ \frac{1}{2} m\left(k v_e\right)^2=\frac{m g h}{1+\frac{h}{R}}$
$ \Rightarrow \frac{1}{2} m k^2 2 g R=\frac{m g h}{1+\frac{h}{R}} $
$ \Rightarrow h=\frac{R k^2}{1-k^2}$
Height of Projectile from the earth's surface $=h$Height from the centre
$r=R+h=R+\frac{R k^2}{1-k^2}$
By solving $r=\frac{R}{1-k^2}$

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