A projectile is projected with velocity kvₑ in vertically upward direction from the ground into the space. ( vₑ 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)

The correct option is R/(1-k²) (which corresponds to Option 3: $\frac{R}{1-k^2}$).

Step-by-Step Explanation:
Let $M$ be the mass of the Earth and $R$ be the radius of the Earth. Let $m$ be the mass of the projectile.
The escape velocity from the surface of the Earth is given by the formula:
$v_e=\sqrt{\frac{2GM}{R}}$

The projectile is projected vertically upward from the surface of the Earth with an initial velocity $u=k{v}_e$, where $k<1$.
Substituting the expression for $v_e$:
$u=k\sqrt{\frac{2GM}{R}}$

Let $r$ be the maximum distance from the center of the Earth reached by the projectile. At this maximum height, the velocity of the projectile becomes zero ($v=0$).

Since air resistance is negligible, the total mechanical energy of the projectile is conserved.
According to the law of conservation of energy:
Total Energy at the Earth's surface = Total Energy at the maximum height
$K_{\mathrm{initial}}+U_{\mathrm{initial}}=K_{\mathrm{final}}+U_{\mathrm{final}}$

Substituting the expressions for kinetic and gravitational potential energy:
$\frac{1}{2}m{u}^2-\frac{GMm}{R}=0-\frac{GMm}{r}$

We can cancel the mass of the projectile $m$ from both sides:
$\frac{1}{2}{u}^2-\frac{GM}{R}=-\frac{GM}{r}$

Now, substitute the value of $u^2=k^2\left(\frac{2GM}{R}\right)$ into the equation:
$\frac{1}{2}\left(k^2\frac{2GM}{R}\right)-\frac{GM}{R}=-\frac{GM}{r}$

Simplify the left side of the equation:
$k^2\frac{GM}{R}-\frac{GM}{R}=-\frac{GM}{r}$

Divide the entire equation by $GM$:
$\frac{k^2}{R}-\frac{1}{R}=-\frac{1}{r}$

Multiply both sides by -1:
$\frac{1}{R}-\frac{k^2}{R}=\frac{1}{r}$

Combine the terms on the left-hand side over a common denominator:
$\frac{1-k^2}{R}=\frac{1}{r}$

Taking the reciprocal of both sides to solve for the maximum distance $r$ from the center of the Earth:
$r=\frac{R}{1-k^2}$