Respuesta :
Answer:
Orbital speed of the satellite is [tex]\frac{\sqrt{Gm_p} }{a}[/tex] .
Explanation:
Given:
Gravitational constant = [tex]G[/tex]
Mass of the satellite = [tex]m_s[/tex]
Mass of the planet = [tex]m_p[/tex]
Radius of the orbit = [tex]a[/tex]
We have to derive the expressions for the orbital speed.
Let the orbital speed be 'vs'.
According to the question:
Force between the planet and the satellite.
From universal law of gravitation.
⇒ [tex]F=\frac{Gm_pm_s}{a^2}[/tex] ...equation (i)
And
Their is centripetal force acting towards the planet.
And we know centripetal acceleration [tex]a_c[/tex] = [tex]\frac{v^2}{r}[/tex] .
From Newtons second law.
⇒ [tex]F=ma[/tex]
⇒ [tex]F=m\frac{v^2}{r}[/tex]
Here the velocity is vs and r = a and mass of the satellite is ms.
⇒ [tex]F=m_s\frac{v_s^2}{a}[/tex] ...equation (ii)
Equating both the equations.
equation (i) = equation (ii)
⇒ [tex]\frac{Gm_pm_s}{a^2} = m_s\frac{v_s^2}{a}[/tex]
⇒ [tex]\frac{Gm_p}{a} =v_s^2[/tex]
⇒ [tex]\sqrt{\frac{Gm_p}{a} } =v_s[/tex]
So,
The orbital speed of the satellite is Sq-rt(Gm_p/a).
