Question

P

P3. A spacecraft speed v 5.12 km/s. [Given: Mass of Earth M 5.98 x10 kg, radius of Earth R 6.37 x 10 m, gravitational constant G 6.67 x101 N m/kg.] of unknown mass is moving on a circular orbit about Earth at a orbit earth (a) Determine the radius r of the circular orbit. |2 points] (b) What is the period T of the orbit? 12 pointsl (c) The satellite, by firing its engines moves to a new circular orbit of radius r' 2r/3. What is the resulting new speed v' of the spacecraft in its new orbit? [2 points] (d) Determine the mass of the If it cannot be determined, explain why. 12 pointsl Spacarap

lease show step by step and unit conversions!

Answer 1

Centripetal force on the spacecraft is given by

$F_{c} = m \, \frac{v^{2}}{r}$

Gravitational force on the spacecraft is

m M ђес"

Where, m is mass of the spacecraft

M is mass of the Earth

r is radius of the orbit

G is universal gravitational constant

Since,

  

or,   

2 GM

or, $v = \sqrt{\frac{GM}{r}}$    ...............................(1)

(a) Putting all the given values, we get

  

6.67 × 10-11 × 5.98 × 1024 51902 "-

or,  

6.67 × 10-n × 5.98 × 10 26214400 24

or,  

r = 1.52 × 107 m

Hence, the radius of circular orbit is 1.52 x 10⁷ m .

(b)

Now from Kepler's third law, period of spacecraft is given by

GM

Putting all the value in this equation, we get

47T2 6.67 × 10-11 × 5.98 × 1024 × (1.52 × 1073

or,  $T^{2} = 4 \pi ^{2} \times 8.83 \times 10^{6}$

or, $T^{2} =348.65 \times 10^{6}$

or, $T =18.67 \times 10^{3} \, \, s$

(c) Given, the new radius of the spacecraft is

  $r' = \frac{2r}{3}= \frac{2 \times 1.52 \times 10^{7}}{3} = 1.01 \times 10^{7} \, \, m$

Putting this value of radius in equation (1), we get

$v = \sqrt{\frac{ 6.67 \times 10^{-11} \times 5.98 \times 10^{24}}{1.0 \times 10^{7}}}$

or,

(d) As both the equations are independent of mass ' m ', we can not determine the mass of the spacecraft.