An astronaut takes a ball of mass m from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is xG Mₑm/(21Rₑ). The value of x is: (take Rₑ = 6370 km)
Options

Question

An astronaut takes a ball of mass m from earth to space. He throws the ball into a circular orbit about earth at an altitude of 318.5 km. From earth's surface to the orbit, the change in total mechanical energy of the ball is xG Mₑm/(21Rₑ). The value of x is: (take Rₑ = 6370 km)

Options

G1 Physics · Accepted Answer

Given: h = 318.5 km Rₑ = 6370 km Therefore: h = Rₑ/20 Orbital radius: r = Rₑ + h = 21Rₑ/20 Initial total mechanical energy at Earth's surface, taking the ball at rest: Eᵢ = -GMₑm/Rₑ Final mechanical energy in circular orbit: Ef = -GMₑm/(2r) = -GMₑm/[2(21Rₑ/20)] = -10GMₑm/(21Rₑ) Change in energy: ΔE = Ef - Eᵢ = -10GMₑm/(21Rₑ) + GMₑm/Rₑ = 11GMₑm/(21Rₑ) Hence: x = 11