Question

A satellite of mass$m$ is launched vertically upward with an initial speed $u$ from the surface of the earth. After it reaches height$R(R=$ radius of earth), it ejects a rocket of mass $\frac{m}{10}$ so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is ($G$= gravitational constant;$M$ is the mass of earth)

• A. $5m\left[u^2-\frac{119}{200}\frac{GM}{R}\right]$
• B. $\frac{m}{20}{\left[u-\sqrt{\frac{2GM}{3R}}\right]}^2$
• C. $\frac{3m}{8}{\left[u+\sqrt{\frac{5GM}{6R}}\right]}^2$
• D. $\frac{m}{20}\left[u^2+\frac{113}{200}\frac{GM}{R}\right]$

Answer 1

The correct option is A

$5m\left[u^2-\frac{119}{200}\frac{GM}{R}\right]$

Step 1. Given data :

Satellite mass $=m$

Speed $=u$

Radius of earth $=R$

Rocket mass $=\frac{m}{10}$

Gravitational constant $=G$

Mass of earth $=M$

Step 2. Find the value of $V$ :

Applying energy conservation

$K_i+U_i=K_f+U_f$

Where, $K_i=$initial kinetic energy

$U_i=$initial potential energy

$K_f=$final kinetic energy

$U_f=$final potential energy

$\frac{1}{2}m{u}^2+\left(-\frac{GMm}{R}\right)=\frac{1}{2}m{v}^2-\frac{GMm}{2R}$

$\Rightarrow$ $v=\sqrt{u^2-\frac{GM}{R}}$ …(1)

By momentum conservation, we have

$m_1{u}_1+m_2{u}_2=m_1{v}_1+m_2{v}_2$

Where,

$m_1$is mass of the first body

$m_2$ is the mass of the second body

$u_1$ and $u_2$ are the initial velocities and

$v_1$ and $v_2$ are the final velocities.

Let the rocket has velocity $V_r$​ in radial direction and $V_T$ tangential direction.

Applying momentum conservation in tangential direction.

$\frac{m}{10}{v}_T=\frac{9m}{10}\sqrt{\frac{GM}{2R}}$ ….(2)

and $\frac{m}{10}{v}_r=mv$

$\Rightarrow$ $\frac{m}{10}{v}_r=m\sqrt{u^2-\frac{GM}{R}}$ …..(3)

Step 3. Find the kinetic energy of the object :

kinetic energy of rocket $=\frac{1}{2}\times \frac{m}{10}({V_T}^2+{V_r}^2)$

$$\begin{gathered} =\frac{m}{20}(81\frac{GM}{2R}+100u^2-100\frac{GM}{R}) \\ =\frac{m}{20}\left(100u^2-\frac{119GM}{2R}\right) \end{gathered}$$

$=5m\left[u^2-\frac{119}{200}\frac{GM}{R}\right]$

Hence, (A) is the correct option.