Radio transmissions show observers in Houston that the International Space Station is 323 miles away, the Chi- nese space statio
n Tiangong is 462 miles away, and the angle International Space Station–Houston–Tiangong is 109 . How far apart are the two space stations?
1 answer:
Answer:
The two space stations are 644.1653 miles far apart.
Explanation:
The distance of Huston and the International Space Station is 323 miles.
The distance of Huston and the Chinese space station Tiangong is 462 miles.
Angle between the two stations = 109⁰
So, applying the law of cosines as;
c² = a² + b² - 2abcosC
From the picture shown below, a = 323 miles and b = 462 miles
So,
c² = 323² + 462² + 2(323)(462)cos109 = 104329 + 213444 - 298452cos109
cos109 = -0.3256
So,
c² = 414948.9712
<u>c = 644.1653 miles</u>
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Answer:
a) K = 2/3 π G m ρ R₁³ / R₂ , b) U = - G m M / r
Explanation:
The law of universal gravitation is
F = G m M / r²
Part A
Let's use Newton's second law
F = m a
The acceleration is centripetal
a = v² / R₂
G m M / R₂² = m v² / R₂
v² = G M / R₂
They give us the density of the planet
ρ = M / V
V = 4/3 π R₁³
M = ρ V
M = ρ 4/3 π R₁³
v² = 4/3 π G ρ R₁³ / R₂
K = ½ m v²
K = ½ m (4/3 π G ρ R₁³ / R₂)
K = 2/3 π G m ρ R₁³ / R₂
Part B
Potential energy and strength are related
F = - dU / dr
∫ dU = - ∫ F. dr
The force was directed towards the center and the vector r outwards therefore there is an angle of 180º between the two cos 180 = -1
U- U₀ = G m M ∫ dr / r²
U - U₀ = G m M (- r⁻¹)
We evaluate for
U - U₀ = -G m M (1 / - 1 /
)
They indicate that for ri = ∞ U₀ = 0
U = - G m M / r