Answer:
v = 200.005 m / s
Explanation:
For this exercise we can use the concept of energy conservation,
Starting point. When the asteroid at h = 1000 km = 1 10⁶ m
Em₀ = K + U = ½ m v₀² - G m M / (R + h)
Final point. When the asteroid is on the surface of the moon
Emf = K + U = ½ m v² - G m M / R
As there is no friction the energy is conserved
Em₀ = Emf
½ m v₀² - G m M / (R + h) = ½ m v² - G m M / R
½ m (v₀² –v²) = G m M (-1 / R + 1 / R + h)
½ (v₀² - v²) = G M (-h / R (R + h))
v² = v₀² + 2 G M h / R (R + h))
Let's calculate
v² = 200² + 2 6.67 10⁻¹¹ 7.36 10²² / 1.74 (1.74 + 1 ))10¹²
v2 = 40000 + 2,059
v = 200.005 m / s