Question

Спутник сечением 1м^2 движется по околоземной орбите на высоте 200км. Определите число соударений молекул воздуха со спутником за 1с. Атмосферное давление на этой высоте 1,37*10^4 Па, а температура 1226К

Answer 1

Поскольку
P = nkT
то концентрация молекул воздуха на этой высоте будет равна
n = P/(kT)
где
P = 1.37·10⁴ Па - давление
T = 1226 K - температура
k = 1.4·10⁻²³ Дж/град - постоянная Больцмана
Количество N молекул в объёме цилиндра с основанием, равным сечению спутника
S = 1 м³
и высотой
Н = vt
где
v - скорость спутника на орбите
t - некоторое время полёта
равно
N = nSvt
Таким образом, количество соударений молекул со спутником в единицу времени будет:
N/t = nSv
Скорость спутника на круговой орбите на высоте
h = 200 000 м
над поверхностью Земли равно:
v = R√(g/(R + h)) (выводить не буду, здесь и так избыток алгебры)
где
R = 6 400 000 м
g = 10 м c⁻²
Таким образом, искомое  количество соударений молекул со спутником в единицу времени составляет:
N/t = SPR√((g/(R + h))/(kT) = 1·1.37·10⁴·6.4·10⁶√((10/(6.6·10⁶))/(1.4·10⁻²³·1226) = 6·10³³ молекул в секунду

Answer 2

Calculation of the Number of Collisions

To determine the number of collisions between air molecules and the satellite in one second, we need to consider the cross-sectional area of the satellite and the number of air molecules that collide with it.

Given: - Cross-sectional area of the satellite: 1 m^2 - Altitude of the satellite: 200 km - Atmospheric pressure at this altitude: 1.37 * 10^4 Pa - Temperature: 1226 K

To calculate the number of collisions, we can use the ideal gas law, which states that the number of gas molecules in a given volume is proportional to the pressure, temperature, and volume of the gas.

The formula for the ideal gas law is:

PV = nRT

Where: - P is the pressure - V is the volume - n is the number of moles of gas - R is the ideal gas constant - T is the temperature

In this case, we are interested in the number of air molecules colliding with the satellite, so we can rearrange the formula to solve for n:

n = PV / RT

To calculate the number of collisions, we need to determine the number of air molecules in a volume equal to the cross-sectional area of the satellite.

The volume can be calculated using the formula:

V = A * h

Where: - A is the cross-sectional area of the satellite - h is the height of the satellite from the Earth's surface

Substituting the values into the formula:

V = 1 m^2 * 200 km

Converting the height from kilometers to meters:

V = 1 m^2 * 200,000 m

Now we can substitute the values into the ideal gas law formula:

n = (1.37 * 10^4 Pa) * (1 m^2 * 200,000 m) / (8.314 J/(mol*K) * 1226 K)

Calculating the result:

n ≈ 2.82 * 10^19 molecules

Therefore, the approximate number of collisions between air molecules and the satellite in one second is 2.82 * 10^19 collisions.

Please note that this calculation assumes an ideal gas behavior and does not take into account factors such as the size and shape of the air molecules or the specific composition of the atmosphere at that altitude. It provides an estimate based on the given information.