Question

The inhabitants of a distant planet, Altair IV, wish to launch their first satellite. The radius of Altair IV is 7.2 x 106 m, and its mass is 1.3 x 1025 kg. The satellite has a mass of 90 kg. Please answer each of the following questions. a) If the satellite is placed in an orbit 400 km above the surface, what is its orbital period? b) What is the kinetic energy of the satellite in this orbit?

Answer 1

a) 1.24 h

b)
`5.132 * 10^(9) J`

Step-by-step explanation:

a) The gravitational force of attraction between satellite and the planet,

which provides centripetal force to keep satellite in circular orbit

`(G M_(p) m)/((R+h)^(2)) &=(m v^(2))/((R+h))`

`v =\sqrt{(G M_(p))/(R+h)}`

`=\sqrt{(\left(6.67 * 10^(-11) N \cdot m ^(2) / kg ^(2)\right)\left(1.3 * 10^(25) kg \right))/(7.2 * 10^(6) m +400 * 10^(3) m )}`

`=1.068 * 10^(4) m / s` .......(i)

The expression for time period of the satellite is,

`T=(2 \pi(R+h))/(v)`

`=(2 \pi\left(7.2 * 10^(6) m +400 * 10^(3) m \right))/(1.068 * 10^(4) m / s )`

`=4468.9 s \left((1.0 h )/(3600 s )\right)`

`=1.24 h`

b) We have from (i) velocity of the satellite as

`v=1.068 * 10^(4) m / s`

Kinetic energy of the satellite is,

`K E &=(1)/(2) m v^(2)`

`=(1)/(2)(90 kg )\left(1.068 * 10^(4) m / s \right)^(2)`

`=5.132 * 10^(9) J`