Question

Два супутники обертаються навколо Землі в одному напрямку по круговим орбітам, причому радіус орбіти другого супутника в 3 рази більший за радіус орбіти першого супутника: r2=3r1. Величина швидкості руху супутника 1 у системі відліку, пов'язаної з центром Землі (але не обертається разом із Землею), дорівнює v1 = 3 км/с. Знайдіть швидкість супутника 2 щодо супутника 1, коли вони знаходяться один над одним на прямій, що проходить через центр Землі.

Answer 1

Відповідь: ≈ 1,27 км/с

Пояснення:

Answer 2

Problem Analysis

We are given that two satellites are orbiting the Earth in the same direction on circular orbits. The radius of the second satellite's orbit is three times larger than the radius of the first satellite's orbit. The velocity of the first satellite in a reference frame centered at the Earth is given as 3 km/s. We need to find the velocity of the second satellite relative to the first satellite when they are aligned on a straight line passing through the center of the Earth.

Solution

To solve this problem, we can use the concept of conservation of angular momentum. The angular momentum of an object in circular motion is given by the product of its moment of inertia and angular velocity. Since the satellites are in circular orbits, their angular momenta are conserved.

Let's denote the moment of inertia of the first satellite as I1, the moment of inertia of the second satellite as I2, the angular velocity of the first satellite as ω1, and the angular velocity of the second satellite as ω2.

Since the satellites are in circular orbits, their moment of inertia can be expressed as the product of their mass (m) and the square of their distance from the center of the Earth (r):

I1 = m * r1^2 I2 = m * r2^2

Since the satellites are in the same direction and their angular momenta are conserved, we have:

I1 * ω1 = I2 * ω2

Substituting the expressions for I1 and I2, we get:

m * r1^2 * ω1 = m * r2^2 * ω2

Cancelling out the mass (m) from both sides, we have:

r1^2 * ω1 = r2^2 * ω2

We can express the angular velocity (ω) as the ratio of linear velocity (v) to the radius (r):

ω = v / r

Substituting the values for ω1 and r1, we have:

v1 / r1 = ω1

Substituting the values for ω2 and r2, we have:

v2 / r2 = ω2

Substituting these expressions into the conservation of angular momentum equation, we get:

v1 / r1 * r1^2 = v2 / r2 * r2^2

Simplifying the equation, we have:

v1 * r1 = v2 * r2

We are given that r2 = 3 * r1 and v1 = 3 km/s. Substituting these values into the equation, we can solve for v2:

3 km/s * r1 = v2 * 3 * r1

Cancelling out r1 from both sides, we get:

3 km/s = v2 * 3

Dividing both sides by 3, we find:

v2 = 1 km/s

Therefore, the velocity of the second satellite relative to the first satellite when they are aligned on a straight line passing through the center of the Earth is 1 km/s.

Conclusion

The velocity of the second satellite relative to the first satellite when they are aligned on a straight line passing through the center of the Earth is 1 km/s.