1.

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. If from the base station, the rocket system of A has been operated, so that its rocket has been fired in forward direction thus reducing the speed of A by 0.5%, then which of the following will happen?

Answer»

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

Solution :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`


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