Question

On July 15, 2004, NASA launched the Aura spacecraft to study the earth's climate and atmosphere. This satellite was injected into an orbit 705 km above the earth's surface, and we shall assume a circular orbit.

(a) How many hours does it take this satellite to make one orbit?
h

(b) How fast (in km/s) is the Aura spacecraft moving?
km/s

Answer 1

Answer:

a). T = 1.64 hr

b). v = 7.503 km/s

Explanation:

Given :

NASA launched the Aura spacecraft to study the earth's climate and atmosphere.

 Height of the satellite orbit from the earth's surface, h = 705 km

                                                                                            = 705000 m

Therefore we know that,

a).Time period of the space craft is

$T = 2\pi \sqrt{\frac{(R+h)^{3}}{G\times M}}$

where, G = Universal Gravitational constant

               = $6.67 \times 10^{-11} N-m^{2}/kg^{2}$

          M = Mass of the earth

               = 5.98 x $10^{24}$ kg

          R = Radius of the earth

             = 6.38 x $10^{6}$ m

∴$T = 2\pi \sqrt{\frac{(R+h)^{3}}{G\times M}}$

 $T = 2\pi \sqrt{\frac{((6.36\times 10^{6})+705000)^{3}}{6.67\times 10^{-11}\times 5398\times 10^{24}}}$

$T = 5933$ s

           = 1.64 hr

Thus, the satellite will take 1.64 hr to make one orbit.

b). We know velocity of the spacecraft is given by

$v=\sqrt{\frac{G\times M}{R+h}}$

$v=\sqrt{\frac{6.67\times 10^{-11}\times 5.98\times 10^{24}}{(6.38\times 10^{6})+705000}}$

v = 7503 m/s

  = 7.503 km/s

Thus, the Aura satellite is moving with velocity v = 7.503 km/s

 

Answer 2

A) The number of hours it takes for the satellite to make one orbit ( h ) = 1.64 hours

b) The speed of the Aura spacecraft = 7.5 Km/s

Given data

Height of satellite Orbit ( h ) = 705 km ≈ 705000 m

A) Determine the time taken in hours for the satellite to make a single orbit

T = $2\pi \sqrt{\frac{(R+h)^3}{GM} }$  -------- ( 1 )

where ; G = 6.67 * 10⁻¹¹ N-m²/kg²,     M ( mass of earth ) = 5.98 * 10²⁴,

R = 6.38 * 10⁶ m ,     h = 705000 m

Insert the values into equation ( 1 )

T = 5933 secs  

   = 1.64 hours   ( time taken to complete one orbit )

B) Determine how fast Aura spacecraft is moving

V = $\sqrt{\frac{GM}{R+h} }$   -------- ( 2 )

where ; G = 6.67 * 10⁻¹¹ N-m²/kg²,     M ( mass of earth ) = 5.98 * 10²⁴,

R = 6.38 * 10⁶ m ,     h = 705000 m

Insert values into equation ( 2 )

∴ V = 7503 m/s

       = 7.5 km/s

Hence we can conclude that the  number of hours it takes for the satellite to make one orbit ( h ) = 1.64 hours and The speed of the Aura spacecraft = 7.5 Km/s

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