Question

A spacecraft is placed in a circular orbit around a planet with mass 6.4 x 1023 kg. The spacecraft orbits at a height of 4.5 x 107 m above the planet’s surface. What additional information is needed to calculate the speed of the spacecraft in the orbit?

Answer 1

Answer:

It is necessary to know the radius of the planet.  

Explanation:

The speed of the spacecraft can be found by means of the equation of the Universal law of gravity:

       

$F = G \frac{M.m}{r^{2}}$  (1)

Where F is the gravitational force, G is gravitational constant, M is the mass of the planet, m is the mass of the spacecraft and r is the orbital radius of the spacecraft.

Equation 1 can be express in terms of the speed by using Newton's second law and the equation for centripetal acceleration:

$F = ma$  (2)

Replacing equation 2 in equation 1 it is gotten:

$ma = G \frac{M.m}{r^{2}}$ (3)

the centripetal acceleration is defined as:

$a = \frac{v^{2}}{r}$  (4)

Replacing equation 4 in equation 3 it is gotten:

$m\frac{v^{2}}{r} = G \frac{M.m}{r^{2}}$ (5)

Then, v can be isolated from equation 5:

$mv^{2} = G \frac{M.m}{r}$

$v^{2} = G \frac{M.m}{rm}$

$v^{2} = G \frac{M}{r}$

$v = \sqrt{\frac{GM}{r}}$

However, the orbital radius of the spacecraft is obtained by the sum of the radius of the planet and the height of the spacecraft above the surface of the planet (r = R+h)

$v = \sqrt{\frac{GM}{R+h}}$  (6)

Hence, by equation 6 it can be concluded that it is necessary to know the radius of the planet in order to calculate the speed of the spacecraft.