Question

Two satellites, A and B, both of mass m = 115 kg, move in the same circular orbit of radius r = 7.57 106 m around Earth but in opposite senses of rotation and therefore on a collision course.

a) What is the total mechanical energy E_A + E_B of the two satellites + Earth system before the collision?

b)If the collision is completely inelastic so that the wreckage remains as one piece of tangled material (mass = 2m), what is the total mechanical energy immediately after the collision?

answer has to be in J

Answer 1

Given that :

mass of the both satellite, m_A = m_B = m = 115 kg

orbital radius, r = 7.57 x 10⁶ m

(a) The total mechanical energy of the two satellites + Earth system before the collision which is given as :

E_A = (1/2) m_A v₀² + (-G M_E m_A / r)

where, v₀ = orbital speed = G M_E / r

E_A = m_A (G M_E / 2r) - (G M_E m_A / r)

E_A = - (G M_E m_A / 2r)                                           

assuming that, m << M_E       

E_A = - (G M_E / 2r)                                                                   { eq.1 }

And

E_B = - (G M_E / 2r)                                                                   { eq.2 }

then, we have E_total = E_A + E_B - (G M_E / 2r) - (G M_E / 2r)

E_A + E_B = - G M_E / r                                                                        { eq.3 }

where, mass of earth, M_E = 5.97 x 10²⁴ kg

G = gravitational constant = 6.67 x 10^-11 Nm²/kg²

inserting the values in eq.3,

E_A + E_B = - (6.67 x 10^-11 Nm²/kg²) (5.97 x 10²⁴ kg) / (7.57 x 10⁶ m)

E_A + E_B = - (39.8 x 10¹³ Nm²) / (7.57 x 10⁶ m)

E_A + E_B = - 5.25 x 10⁷ J

E_total = - 5.25 x 10⁷ J

(b) If the collision is completely inelastic so that the wreckage remains as one piece of tangled material (mass = 2m), then the total mechanical energy immediately after the collision which is given as :

since, v = 0     (it means that kinetic energy is zero)

then we have,

E_total = - G M_E 2m / r                                                                       { eq.4 }

inserting the values in eq.2,

E_total = - (6.67 x 10^-11 Nm²/kg²) (5.97 x 10²⁴ kg) 2 (115 kg) / (7.57 x 10⁶ m)

E_total = - (9158.5 x 10¹³ Nm²) / (7.57 x 10⁶ m)

E_total = - 1209.8 x 10⁷ J

E_total = - 1.21 x 10⁴ J