Respuesta :
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
The asteroid moves from a high elevation to a lower one therefore potential energy is converted to potential energy.
The speed of the meteorite when it impacts the lunar surface is approximately 1,449.9 m/s
Reasons:
The Kinetic and potential energies are;
Using the energy conservation equation, we get;
[tex]-\dfrac{G \cdot M \cdot m}{R + h} + \dfrac{1}{2} \cdot m \cdot v_1^2 = -\dfrac{G \cdot M \cdot m}{R } + \dfrac{1}{2} \cdot m \cdot v_2^2[/tex]
Where;
G = Universal gravitation constant = 6.67408× 10⁻¹¹ m³/(kg·s²)
M = Mass of the Moon = 7.34767309 × 10²² kg
m = Mass of the asteroid = 100 kg
R = Radius of the Moon = 1,737.400 meters
Therefore, we get;
[tex]\dfrac{G \cdot M \cdot m}{R } -\dfrac{G \cdot M \cdot m}{R + h} + \dfrac{1}{2} \cdot m \cdot v_1^2 = \dfrac{1}{2} \cdot m \cdot v_2^2[/tex]
[tex]G \cdot M \cdot m\cdot \left(\dfrac{1}{R } -\dfrac{1}{R + h} \right)+ \dfrac{1}{2} \cdot m \cdot v_1^2 = \dfrac{1}{2} \cdot m \cdot v_2^2[/tex]
Dividing by m, gives;
[tex]G \cdot M \cdot \left(\dfrac{1}{R } -\dfrac{1}{R + h} \right)+ \dfrac{1}{2} \cdot v_1^2 = \dfrac{1}{2} \cdot v_2^2[/tex]
Plugging in the values gives;
[tex]6.67408 \times 10^{-11} \times 7.34767309 \times 10^ {22}\left(\dfrac{1}{1737400 } - \dfrac{1}{1737400 + 1 \times 10^6} \right)+ \dfrac{1}{2} \times 200^2 = \dfrac{1}{2} \cdot v_2^2[/tex]
Which gives;
[tex]1051105.6191 = \dfrac{1}{2} \cdot v_2^2[/tex]
v₂² = 2× 1051105.6191 = 2102211.2382
v₂ = √(2102211.2382) ≈ 1,449.9
- The speed of the meteorite when it impacts the lunar surface, v₂ ≈ 1,449.9 m/s
Learn more here:
https://brainly.com/question/20614605
