An unmanned satellite `A` and a spacecraft `B` are orbiting around the earth in the same circular orbit as shown.
The spacecraft is ahead of the satellite by some time. Let us consider that some technical problem has arisen in the satellite and the astronaut from `B` has made it correct. For this to be done docking of two (`A` and `B`) is required (in layman terms connecting `A` and `B`). To achieve this, the rockets of `A` have been fired in forward direction and docking takes place as shown in the figure below:

Take mass of the earth `=5.98xx10^(24)kg`
Radius of the earth `=6400 km`
Orbital radius `=9600km`
Mass of satellite `A=320 kg`
Mass of spacecraft `=3200 kg`
Assume that initially spacecraft `B` leads satellite `A` by `100s`, i.e., `A` arrives at any particle position after `100 s` of `B’s` arrival. Based on the above information answer the following questions
The initial total energy and time period of satellite are, respectively,

Options

A. Its orbit becomes elliptical with semi-major axis `9505.3 km`.
B. its total energy becomes `-6.714xx10^(9)J`
C. Its new time period becomes `9219.67s`.
D. All of the above.

Answer

D

Explanation

Speed of a before firing the rocket,
`v_(i)=sqrt(GM//r)=644.582ms^(-1)`
After firing the rocket speed of `A` is
`V_(f)=0.995, v_(i)=6413.6ms^(-1)`
Energy of the satellite after firing,
`E_(f)=2E+(mv_(r)^(2))/w=-6.714xx10^(9)J`
as due to firing potential energy doens not change.
From `E=-(GMm)/(2a),` the new semi major axis is given by
`a=-(GMm)/(2E_(f))impliesa=9505.3km`
New time period `T'=(2pia^(32//3))/(sqrt(GM))=9219.67s`