A meteoroid is moving towards a planet. It has mass m= 0.18x10⁹ kg and speed v1 = 5.9x10⁷ m/s at distance R1 = 2.9x10⁷ m from the center of the planet. The radius of the planet is R = 0.22x10⁷ m. The mass of the planet is M = 2.4x10²⁵ kg. There is no air around the planet.
Calculate the speed of the meteoroid just prior to the instant when it hits the surface of the planet, in meters per second.

5.90000114e+7 m/s

Explanation:

Given values

M₁ = 1.8e+8 kg

M₂ = 2.4e+25 kg

M = M₁ + M₂ = 2.400000000000000018e+25 kg

G = 6.6743e-11 m³ kg⁻¹ sec⁻²

GM = 1.60183200000000001201374e+15 m³ sec⁻²

r₁ = 2.9e+7 m

v₁ = 5.9e+7 m/s

R = 2.2e+6 m

The total energy of the meteroid per unit mass at time t₁ is

E/M₁ = 0.5v₁² − GM/r₁ = 1.7404999447644137931034478615952e+15 m²/s²

At impact,

E/M₁ = 0.5v₂² − GM/R = 1.7404999447644137931034478615952e+15 m²/s²

v₂ = √[2(E/M₁ + GM/R)] = 5.9000011404572937396882314817886e+7 m/s

v₂ − v₁ ≈ 11.4 m/s