1. The physics of Mars and its two moons.
Mars has two moons, Phobos and Deimos (Fear and Panic, the companions of Mars, the god of war). Deimos has a period of 30 h, 18 min and a mean distance from the centre of Mars of 2.3 x 104 km. If the period of Phobos is 7 h, 39 min, what is the mean distance (in km) from the centre of Mars?
9.2
F = m a
G m Mmars/R^2 = m V^2/R
V^2 R = G Mmars
V = (G Mmars)^.5 R^-.5
period = 2 pi R/V
period = 2 pi R/(G Mmars)^.5 R^-.5
period = T = constant R^1.5 = k R^1.5
T1 = k R1^1.5
T2 = k R2^1.5
T2/T1 = R2^1.5/R1^1.5
so R2^1.5 = R1^1.5 (T2/T1)
To find the mean distance of Phobos from the center of Mars, we can use Kepler's third law. This law states that the square of the period of a moon is proportional to the cube of its mean distance from the center of its planet.
Let's denote the period of Phobos as T1 and the mean distance from the center of Mars as d1. We'll also use T2 and d2 to represent the corresponding values for Deimos.
According to Kepler's third law:
(T1/T2)^2 = (d1/d2)^3
Plugging in the values we know:
(T1/30 hours 18 minutes)^2 = (d1/2.3 x 10^4 km)^3
(7 hours 39 minutes/30 hours 18 minutes)^2 = (d1/2.3 x 10^4 km)^3
To simplify the calculation, let's convert the periods to just minutes:
(459 minutes/1818 minutes)^2 = (d1/2.3 x 10^4 km)^3
0.252^2 = (d1/2.3 x 10^4 km)^3
0.063504 = d1^3/(2.3 x 10^4 km)^3
Let's solve for d1 by taking the cube root of both sides:
(0.063504)^(1/3) = d1/(2.3 x 10^4 km)
0.400488 = d1/(2.3 x 10^4 km)
Now, we can solve for d1 by multiplying both sides by (2.3 x 10^4 km):
0.400488 * (2.3 x 10^4 km) = d1
9208.152 km = d1
Therefore, the mean distance of Phobos from the center of Mars is approximately 9208.152 km.
To answer this question, we can use Kepler's third law, also known as the Law of Periods. This law states that the square of the period of an orbiting object is proportional to the cube of its average distance from the center of the object it is orbiting.
We have the period of Deimos, which is 30 hours and 18 minutes. To convert this to minutes, we multiply 30 by 60 (to convert hours to minutes), and then add 18. So the period of Deimos is (30 x 60) + 18 = 1818 minutes.
Using the information given, the period of Phobos is 7 hours and 39 minutes, which is (7 x 60) + 39 = 459 minutes.
Next, we can set up a ratio using the Law of Periods. Let T1 be the period of Phobos in minutes, T2 be the period of Deimos in minutes, D1 be the distance of Phobos from the center of Mars, and D2 be the distance of Deimos from the center of Mars.
(T1/T2)^2 = (D1/D2)^3
Plugging in the values we have:
(459/1818)^2 = (D1/2.3 x 10^4)^3
Simplifying further:
(0.2523)^2 = (D1/(2.3 x 10^4))^3
0.0636 = D1^3/(2.3 x 10^4)^3
Raising both sides to the power of 1/3:
(0.0636)^(1/3) = D1/(2.3 x 10^4)
Simplifying:
0.4064 = D1/(2.3 x 10^4)
Multiplying both sides by (2.3 x 10^4):
D1 = 0.4064 x (2.3 x 10^4)
D1 ≈ 9343.2 km
Therefore, the mean distance from the center of Mars for Phobos is approximately 9343.2 km.